
h'\\\'\ 



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! Ul M.I ! Tlln "il llM I'l 



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/|l^t|l||!|f^^ 





Book ,L 5E 

iis!u>i° 



COPYRIGHT DEPOSrr. 





O'REILLY ESTATE BUILDING, ST. LOUIS, MO. 

A. B. Groves, Architect IMurch Bros. Construction Co., Contractors 

G. S. Bergendahl, M. AM., Soc. C E., Engineer, St. Louis Representative, ^lushroom System 



THE THEORY OF THE FLEXURE 
AND STRENGTH 

OF 

RECTANGULAR FLAT 
PLATES 

APPLIED TO 

REINFORCED CONCRETE 
FLOOR SLABS 



BY 



HENRY T. EDDY, C. E., Ph. D., Sc. D. 

PROFESSOR OF MATHEMATICS AND MECHANICS, COLLEGE OF ENGINEERING, 

AND DEAN OF THE GRADUATE SCHOOL, EMERITUS, UNIVERSITY OF MINNESOTA. 

CHAIRMAN OF BOARD OF EXAMINERS FOR THE MUSHROOM SYSTEM 

APPOINTED BY THE COMMISSIONER OF BUILDINGS 

CHICAGO, JUNE 15, 1912 



PRICE ^5.00 



lU^inKl^S it COMPANY, MINNEAPOLIS 
1913 



\ f\ 4A-5 
E 3 ^ 



COPYRIGHT 1913 
BY 

HEXRY T.EDDY 

ALL EIGHTS RESERVED 



0r^ 

©CI.A34-;698 



PREFACE 

In reviewing the history of every type of structural work, we find 
the designing engineer influenced in his first attempts at any new 
type of structure by his knowledge of the practical forms of con- 
struction with which he is already familiar. In fact until very 
recent times precedent was the engineer's sole guide. It was to be 
expected therefore that the pioneers in concrete-steel construction 
should follow and closely imitate timber and structural steel con- 
struction. In following this type, the idea has been to build up the 
structure as a whole by assembling and joining together a number 
of independent elements or units: whereas concrete, with or without 
reinforcement, is a kind of material that is best suited by its nature 
to construction in monolithic form. But when the attempt has been 
made to treat such structures theoretically, preconceived ideas 
have led to the effort to treat them by analysing them into separate 
members and computing the strength of these arbitrarily selected 
units, assumed to act independently as they do in steel structures. 
Such treatment has led to errors of as much as two or three hundred 
percent in the computation of slabs with two way reinforcement 
supported on four sides, and to errors of four hundred percent in 
case of continuous flat slab construction such as occurs in the mush- 
room system. 

When we consider the fact that fire losses in Canada and the 
United States amount each year to half a billion dollars, and that 
the question of commercial economy determines whether buildings 
shall be built of fireproof and incombustible materials such as rein- 
forced concrete, or of inflammable materials such as are used in timber 
construction, it is at once evident how important it is to the general 
public to be able to determine on theoretically correct principles 
whether safe fin^proof buildings can be built at approximately the 
same or less cost than combustible ones. In case of any uncertainty 
on this question, the designer is compelled for safety to (Mn])l()y 
materials in such lavish amounts as to render cost prohibitive. 

The failure of engineers and mathematicians generally to n])|)ly 
the matliematical th(M)ry of elasticity to the new material concrt^te- 
steel, has led to (considerable controversy betwcHMi ])ra('ti('al con- 
structors of experience, ami theoretical engiiuHM-s without sucli 
exp(>rienc(\ 



IV PREFACE 

Marsh in his treatise on Reinforced Concrete, edition of 1905, 
Part V, p. 209, makes the following remarks upon this subject: — • 

When properly combined with metal, concrete appears to gain 
properties which do not exist in the material when by itself, and 
although much has been done b}' various experimenters in recent 
years to increase our knowledge on the subject of the elastic be- 
haviour of reinforced concrete, we are still very far from having 
a tiTie perception of the characteristics of the composite material. 

It may be that we are wrong from the commencement in 
attempting to treat it after the manner of structural ironwork, and 
that although the proper allowances for the elastic properties of the 
dual material is an advancement on the empirical formulae at first 
emploA'ed, and used by many constructors at the present time, yet 
we maj be entirely wrong in our method of treatment. 

The molecular theory, i. e. the prevention of molecular defor- 
mation by supplying resistances of the reverse kind to the stresses 
on small particles, may prove to be the tiTie method of treatment for 
a composite material such as concrete and metal. This theory is 
the basis of the Cottacin construction which certainly produces good 
results and very light structures, and 2vl. Considere's latest researches 
on the subject of hooped concrete are somewhat on these lines.'' 

In this statement, ]\Iarsh undoubtedly has in mind the great 
discrepancy between the results of tests and of computations of 
multiple way reinforcement. 

The empirical formulas of Hennebique, a pioneer in this field 
and one of its most extensive investigators, give numerical results 
at variance with the usual published theories for two waj^ reinforce- 
ment, while the empirical formulas published by Turner in 1908 
exhibit an equally radical divergence. 

It has remained apparently for Dr. Eddy to discover the reason 
for this great discrepancy by a rigid application of the mathematical 
theory of elasticity to the problem presented by multiple way rein- 
forcement. 

The clearing up of the mathematical difficulties with which the 
theoretical engineer has heretofore struggled in dealing with 
remforced concrete, will lead to its more general adoption by the 
elimination of uncertainty ui design; and will lead to the adoption 
of those types which are safest to erect, and those which possess a 
degree of toughness, due to their monolithic construction, which is 
lacking in tA^pes that merely imitate older forms of timber and steel 
construction. 

The record of the ^Mushroom s\^stem in the successful construc- 
tion of between one and two thousand buildings, without accident 
to the workmen, and without failure to make good the guarantee 



PREFACE 



of test capacity, can be accounted for only on the ground that 
scientifically designed multiple way reinforcement is inherently safer 
to erect, and more reliable at all stages of construction than other 
types. The theory given in detail which accounts for its economy 
and safety, should, we believe be of interest to the profession at 
large, and should prove of value to the practicing engineer in check- 
ing designs for his clients. 

The endorsement of this theory by the undersigned does not 
constitute a license to use the patented type, though the extension 
of the theory to older types which are not patented will undoubtedly^ 
lead to the closest competition with it along legitimate lines. Any 
loss to the patentee arising from such competition, will, in the writer's 
judgment, be more than counterbalanced from a commercial stand- 
point, by the increased safety and corresponding popularity of 
reinforced concrete as a material of construction. 

The invention covered by the claims of the broad patent 
just mentioned includes much more than merely the standard mush- 
room system. This is, however, the one form of all others 
of the patent, which the writer prefers by reason of certain practi- 
cal advantages which it possesses. Its arrangement of parts whereby 
a continous- multiple way reinforced flat slab is supported by a 
large cantilever column head integral with the column and em- 
bedded in the slab so as to resist tensile stresses both radial 
and circumferential in zones near the top of the slab over 
and around the columns, and in the zones near the bottom of the 
slab toward the center of the panels, will be fully discussed in succeed- 
ing pages. This discussion, which deals with the mushroom sys- 
tem primarily, is intended as an advance chapter or two of a more 
comprehensive treatise on Concrete-Steel Construction, in wliich 
it is intended to treat somewhat fully concrete columns, beam and 
slab construction, wall panels, etc., as well as flat slabs, and to 
introduce the results of experimental work now undc^- way to 
determine the value of Poisson's ratio for differcMit coinbinations 
of st(^el and concrete. 

This treatise will then re])resent the joint efforts of a ])ro- 
fessional mathematician accustonuHl to treating tlu^se probliMus, 
and a ])i"()iessional builder and designei" of iH^inforciHl conci-iit" with 
many years of ])ractical exixM-ieiice behind him. 

Th(^ pri(;e charg(Ml for this booklist will be cRHlitcvl in I'tMurn 
for it, on the larger treat is(> which \\w authoi's intcMid to coniphMc^ 
as soon as the magnitude of {\w task will i)(M'mit. 

('. A. V. riKM-.K. 



CONTENTS 



Section Page 

1. Flat Slab Floors. The Mushroom System 1-4 

2. Notation 5-7 

3. True and Apparent Bending Moments 8-10 

4. Poisson's Ratio 11-12 

5. General Differential Equation of Moments 13-15 

6. General Differential Equation of Deflections 16-17 

7. Solution of the Differential Equation in Case of Uni- 

form Slab Supported on Columns 17-20 

8. Solution for Side Belts 21-24 

9. Practical Formulas for Stresses in Side Belts 24-30 

10. Practical Formulas for Stresses in Column Heads 31-35 

11. Practical Formulas for Stresses in the Middle Area of 

Panel 36-40 

12. Deflections at Mid Span of the Side and Diagonal Belts. 41-43 

13. Proportionate Deflections at Mid Span and Center of 

Panel 43-46 

14. Radial and Ring Rods, and Shear around Cap 47-55 

15. Standard Mushroom System and other Systems 55-61 

16. Specimen Computation of Slab 62-70 

17. Comparative Test to Destruction of Two Slabs 71-93 

18. Suggestions as to Construction, Finish, Etc 94 100 

Appendix. Specifications for Floors 101 104 



VII 




TISCHERS CREEK BRIDGE, DULUTH, MINN. 
Spans are 26 feet longtitudinally 
This type is built with spans up to 50'0" 




View of Reinforcement in Place 

TISCHERS CREEK BRIDGE, DULUTH, MINN. 

Designed by C. A. P. Turner Geo. H. Lounsbury, Contractor 



FLAT SLAB FLOORS 

1 . The superiority of flat slab floors supported directly on columns, 
over other forms of construction when looked at from the stand- 
point of lower cost, better lighting, greater neatness of appearance, 
and increased safety and rapidity of construction, is so general!}^, or 
rather so universally conceded as to render any reliable information 
relative to the scientific computation of stresses in this type of con- 
struction of great interest. Heidenreich, in his Engineer's Pocket 
Book on Reinforced Concrete, page 89, classifies this type as floors 
without beams and girders — '^Mushroom System." 

Since '^mushroom," as applied to concrete, is an arbitrary or 
fanciful term, and indeed, almost a contradictory one, a word of 
explanation as to its origin may be of interest. The term was 
originated by C. A. P. Turner, of Minneapolis, and applied to his 
flat plate construction, more particularly because of the fancied 
resemblance to the mushroom, of the column and column head 
reinforcement of that particular form of his flat plate construction 
which he seemed to prefer by reason of certain practical advantages. 
Another fancied resemblance is the rapidity of erection, comparable 
to the over-night growth of the mushroom. Here the resemblance 
ceases, since the construction, once erected, is enduring and per- 
manent. 

The Mushroom System is a continuous flat plate of concrete 
supported directly on columns, and reinforced in such a maner that 
circular and radial tensile stresses concentric with the column are 
provided for by metal reinforcement in the tension zone abov(^ the 
columns, and similar provision is made for tensile stresses in the 
lower portion of the slab concentric with the center of the panel, 
diagonally between the columns. Since all forces in a plane may bo 
resolved into equivalent com])onents along any pair of axes at right 
angles to each other, it is possible to provide reinforcement to resist 
any horizontal tensile stresses in the slab by various arraugcMiuMits 
of intersecting belts of rods at zones wIkmv those stresses occiii-. 
All arrangements of this kind are by no in(>ans (Hiunlly i^fT(H'ti\(\ 
A system of wide reinforcing bolts from cokunn to ooluinu ooni- 
bined with a system of radial and ring rods to constitutt^ a lai-g(\ 
substantial cantilever mushroom lu^ad at tlu^ to]) of each colunni 




Fig. 1. Vertical Section of Standard ^Mushroom Head showing posi- 
tion of Radial and Ring Rods, and Slab Rods, Vertical and Hori- 
zontal Sections of Spirally Hooped Column, with Plain Bar Hoop 
Collar Band, Vertical Reinforcing Rods and Elbow Rods. 




Fig. 2. Plan of Reinforcement in Standard ^Mushroom System. 
Radial and Ring Rods, Collar Band and Slab Rods. Diameter of 

Head = ^ = 7/l6(a+6). 



STANDARD MUSHROOM SYSTEM 



provides a very effective and economical arrangement for controlling 
the distribution of the stresses in the slab, and furnishes the resis- 
tance necessary to support these stresses by placing the steel where it 
is most needed. It not only has the same kind of advantage that the 
continuous cantilever beam has over the simple girder for long 
spans, but combines with it the kind of superiority that the dome has 
over the simple arch by reason of circumferential stresses called into 
play, which adds greatly to the carrying capacity of the slab. 

In the standard mushroom type, which is quite fully discussed 
in this paper, the heavy frame work, concentric with the colunrn, 
supports the slab reinforcement at a fixed elevation, furnishes a high 
degree of resistance to shear, and secures a high degree of safety 
during construction. It extends as a cantilever approximately one 
fourth of the way to the next column as shown in Figs. 1 and 2 on 
page 2. Arranged upon the radial rods of this frame rest two or 
more large hoops and upon these rest the wide spreading belts of rods 
which extend both directly and diagonally from column to column. 
Over the columns these belts lie near the upper surface of the slab, but 
they run near the lower surface as they approach points midway 
between columns. 

The cantilever slab thus formed, not only has the same advant- 
ages for this form of construction that the cantilever construction 
has for long span bridges, but it causes the slab to have greater 
stiffness and gives it greater resistance to shear in the neighborhood 
of the columns; it removes the locus of zero bending moment to a 
much greater distance from the column than would otherwise be the 
case, thus dimininishing the area of that part of the slab which tends 
to become concave on its upper face and enlarging the convex area. 

The cantilever frame-work further, not only moves the locus of 
zero bending outward from the column, but it also fixes the locus of 
zero bending moment at a known position so that it does not vary 
with increase and decrease of the load or change of the load from one 
span to an adjacent span as would be the case were the mass of 
metal in the frame and its stiffness largely reduced. This is ac- 
complished as follows: 

Th(; locus of zero bending moments is fixed by the di]) of (lie 
reinforceing rods as they leave the up]KM- surface^ of the slab uviw 
the edge of mushroom and pass Ix^low the ncnitral surfact^ to a \c\c\ 
near the l)ott()m of the slab: Sucli change of tensile resistanct^ in 
the slab ne('(\ssarily l()caliz(\s at tliesc^ ])()ints tlu^ zcvo bcMuling mo- 
ments. 



4 SLAB ACTION BIBLIOGRAPHY 

In addition to the advantages just mentioned, which are 
of so self-evident a character as to be readily appreciated even 
by the layman, there is another of such an obscure and apparently 
inexplicable a nature that it was for years denied as incredible and 
regarded as non-existent by practical builders, and engineers as 
well, unless they had opportunity to be convinced of its reality 
by experiment. I refer here to the additional strength and stiff- 
ness which is imparted to a belt of rods in a given direction in a 
slab by another belt at right angles to the first belt, or at various 
angles with it. This should be designated as slab action proper 
in distinction from cantilever action. It depends for its amount 
upon the value of Poisson's ratio of lateral contraction to direct 
elongation in the slab, and is the basis of the so called circum- 
ferential stresses, which make the strength and stiffness of such 
reinforced flat slabs much greater than they are estimated to be 
when these are neglected, as they usually have been. This mis- 
taken view has in the past constituted the most serious obstacle 
to the adoption of this form of structure, and has been the ground 
of conscientious opposition to its introduction on the part of con- 
sulting engineers. It is the object of this investigation to remove 
if possible all reasonable uncertainty as to the rational theory of 
this form of structure. 

The following partial bibliography of this subject may be useful 
to those unfamiliar with what has been done in this field. 

Concrete Steel Construction, (305pp) 

By C. A. P. Turner, M. Am. Soc. C. E. 
816 Phoenix Bldg., Minneapolis, 1909. 

Reinforced Concrete Construction, (259pp) 

By Turneaure and Maurer, University of Wisconsin 
Wiley, N. Y., 1907. 

Concrete, Plain and Reinforced, (483pp) 
By Taylor and Thompson, 
Wiley, N. Y., 1911. 

Trans. Am. Soc. C. E. 
Vol. LVI. June 1906. 

Engineering News: — ^ 

Oct. 4, 1906, p. 361. 
Feb. 18, 1909, p 176. 
Dec. 23, 1909, p. 694. 

Engineering Record: — 

March 28, 1908, p 374. 
May 2, 1908, p 575. 
Oct. 10, 1909, p 411. 
April 3, 1909, p 408. 
April 10 ,1909, p 492. 



NOTATION 

2. All lengths and areas are measured in inches, and all weights 
in pounds. 
A = area of cross section of steel reinforcement per unit width of 

slab, in case it be assumed to be replaced by a uniform sheet 

of equal weight. 

Ai = area of cross section of all the rods in one side belt. 

A2 = area of cross section of all the rods in one diagonal belt. 

a = one half the longer side of a panel from center to center of 
columns. 

b = one half the shorter side of a panel. 

B = the shortest distance along one side of a panel from the edge 
of a column cap to the edge of the next cap. 

Ci and C2 are constants depending on the relative lengths of the sides 
of any panel, which reduce to unity for any square panel. 

Di = the deflection of the middle of the longer side of the panel 
below the edge of the cap. 

D2 = the deflection of the center of the panel below the edge of the 
cap. 

d = the effective thickness of the slab at any point, being the 
vertical distance from the center of action of the reinforce- 
ment to the compressed surface of the concrete. 

di = the vertical distance from the center of the rods in the side 
belt at mid span to the top surface of the concrete. 

d2 — the distance at the center of the panel from th(^ ('(^iter of the 
rods in the second or upper diagonal l)elt to the top of thc^ 
concrete. 

d:i = the distance at the edge of the cap from tlie center of th(^ tliird 
belt of rods from the top, to the c()m}:)r(^ssod surfact^ of \\\v 
concrete. 

E or K^ = Young's moduhis foi- st(H>l = 3 x 10'. 

E^, = Young's moduhis for concrete. 

Ci = (^longation in stcn^l parallel to long sldv \)v\{. 

62 = elongation in steel ))arallel to shoii side belt. 

ei = elongation in steel parallel to diagonal l)(>lt. 



t) NOTATION 

F = modulus of elastic resistance to shearing. 
i\ = Ee = intensity of actual stress in steel, 
/c = intensity of stress in concrete. 
^ = 7 16 (a+h) = the diameter of the mushroom head and width 

of belts. 
Ji = the total actual thickness of concrete slab. 

id = vertical distance from center of tension of steel to neutral 
surface of slab. 

jd = vertical distance from center of tension in steel to center of 
compression in concret^e. 

I'd = vertical distance from neutral surface to compressed surface 
of concrete, hence i -r I' = 1. 

K = Poisson's ratio of lateral contraction to longituchnal stretch- 
ing for reinforced concrete slabs. 

Zi = 2a = long side of panel between column centers. 

Lo = 2h = short side of panel between column centers. 

I = cUstance from collar band at top of coliman to edge of cap. 

rrii = true moment of resistance of the tensile stresses in steel parallel 
to the long side per unit of width of slab. 

rm = true moment of resistance of steel parallel to short side per 
unit of width. 

mi and mo = apparent moments per unit of width of forces appUed 
parallel to the long and short sides respectively. 

n = the apparent moments per unit of width of the equal 

twisting couples parallel to either side. 

Pi = intensity of the forces applied parallel to the long side. 

po = ditto for short side. 

p = intensity of stress in extreme fiber of radial rods. 

q = load on slab in pounds per square inch. 

J?i and Ro = the radii of curvature of vertical sections of the slab 
parallel to the long and short sides respectively. 

-Si and S2 = the vertical shearing stresses per unit of width of slab 
respectively perpenchcular to the long and short sides 
of the slab. 

^ = the intensity of vertical shearing stress in radial 

rods. 

i; = either of the equal horizontal tangential or shearing 

stresses parallel to the sides of the panel. 



NOTATION. SQUARE PANEL 



t 

u and V 

V 

X y z 



Az 

Zi and Z2 

d 
dz 

dx 



= the thickness of a radial rod. 

= deformations parallel to the long and short sides re- 
spectively. 

= total vertical shearing stress in radial rod. 

= horizontal and vertical coordinates parallel to sides 
of panel. 

= difference of two vertical coordinates. 

= deflections of radial rods. 

= sign of partial differential. 

= partial differential coefficient of z with respect to x. 




Fig. 3. Phui of Reiniorccnient Muslirooni System. 
Square Panel, g = lL (as drawn). 
Line of Ultimate Weakness. 



8 TRUE AND APPARENT STRESSES 

3. As preliminary to a general investigation of the rational 
analysis of the flat slab, it seems desirable in the first place to 
make a brief exposition of the relationship between the true bend- 
ing moments and the apparent bending moments in the flat slab as 
follows: 

The fundamental equations of extensional stress and strain in 
thin flat plates and slabs, established a generation ago and accepted 
by Grashof*and by all authorities on the subject since then, maybe 
written in the forms : 

Eei = pi— Kp2 \ (^-^y 

Ee2 = V2 — Kpi ) 



{1-K')pi =E{e,+Ke2) [ Ha) 

{l—K')p2=Eie2 + Ke,)i 

in which pi and p2 are the external applied or apparent stresses per 
unit of area of cross section of the plate, or of the reinforced slab, 
which act parallel to the axes of x and y respectively if these latter 
lie in the neutral plane of the slab; and ei and 62 are extensometer 
elongations of plate or slab reinforcement per unit of length parallel 
to X and y respectively. E is Young's modulus of elasticity, and K 
is Poisson's ratio of lateral contraction to linear elongation. Any 
piece of material which is subjected to stress, and is of such shape 
that more than one of its dimensions is considerable, as compared 
with its remaining dimension, must have its stresses and strains 
considered with reference to lateral contraction. This is the case 
in plates and slabs, as it is not in case of rods and beams. 

In the above equations Eei and Ee2 are the true stresses per square 
inch of section of reinforcement acting along lines parallel to x and y 
respectively, whatever pi and p2 may be. These latter are the cause 
of true stresses, but are not themselves the values of the true stresses, 
as they are in case of rods, etc., where one dimension only is large. 

These equations show that the elongation ei in the direction of 
X is not dependent alone upon the tension pi applied in that direc- 
tion, for it is diminished by any tension acting along y, but is in- 
creased by any compression acting along y. It thus appears that 
any tension p2 along y assists the piece in resisting elongation along 
x and makes it able to endure safely a larger applied stress pi with 
the same degree of safety, i. e., with the same percentages of elonga- 
tion or true stress. But it is also equally true that any compression 
of amount p2 reduces the safe value of pi which may be applied to 

*Theorie der Elasticitat und Festgkeit, F. Grashof Berlin 1878. 



TRUE AND APPARENT MOMENTS 



it. These principles are not in accordance with those which hold 
in ordinary computations for rods and bars, whose lateral dimensions 
are small compared with their lengths, and whose lateral stresses 
are negligible. This divergence between the true stresses as shown 
by actual deformations, and the apparent or applied stresses, is a 
fruitful source of error in the attempted computation of slabs. 

Equations (1) in their present form apply to simple extensional 
or compressive stresses and strains but may be extended to apply 
to bending of slabs in the following manner: 

Take A as the cross section of the reinforcement per unit of 
width of slab when the actual reinforcement is regarded as distrib- 
uted into a thin sheet of uniform thickness, and let jd be the vertical 
distance from the center of the reinforcement to the center of com- 
pressional resistance of the concrete regarded as a fraction j of d, 
d being the distance from the center of the steel to the top of the 
slab. Then 

mi = Api jd, and m2 = Ap2 jd, (2) 

are the apparent bending moments per unit of width of slab, of the 
applied apparent stresses pi and p2, tending when positive, to cause 
lines which before bending are straight and parallel to x and y re- 
spectively, to become concave upwards. 

Again mi = Eei Ajd, and m2 = Ee2 Aid, (3) 

are the true bending moments of the actual resistance stresses in 
the reinforcement per unit of width of slab, as shown by extenso- 
meter strains in the steel parallel to the axes of x and y respectively. 

Multiply equations (1) thru by Ajd and substitute the values 
given in equations (2) and (3), from which we obtain the following 
relations between the true and apparent bending moments in the slab. 

w 

(-t«) 



mi — lUi — ■ i^m^ 
m2 = m2 — ^^nii J 

(1— /^^)mi = mi + Km2 
(1 — K^)m2 = m2 + Kmi 

These equations bring out in a striking niauner the essential di\(M-- 
gence of the correct theory of slab action from that of beam act ion 
in which latter case we have the well known equations 

mi = mi, and mo = m^ 

i. e., in beams tiie monuMit of tlie ai)pH(Hl forces is (Miiial to tiu* 
moment of tlie internal resistance, which is not true in slabs. 



10 TRUE AND APPARENT MOMENTS 

All attempts to base computations of the deflection of slabs 
upon beam action are therefore necessarily erroneous. Such com- 
putations are inapplicable and misleading, hence deflections and 
stresses in slabs cannot be correctly computed by any form of 
simple or compound beam theory. 

Equations (4) show: 

1st That at points where mi and m2 are of the same sign, (as 
for example in the convex part of the mushroom near the columns 
and also near the center of the panel) the true bending moments 
mi and m2, which determine the actual stresses in the reinforcement 
are less than the apparent bending moments, which latter have been 
ordinarily assumed, according to the beam theory, to determine 
those stresses. 

2nd That the compressive stresses in the concrete around the 
column cap are determined on the same principles as the tensile 
stresses and are consequently reduced in accordance with the value 
of iT by a considerable percentage below values corresponding to 
mi and m2 of the beam theory. 

3rd That at points where mi and m2 have different signs, as 
they have for example in the middle part of the span at the side of 
the panel directly between mushroom heads, the values of the true 
bending moments are larger than the apparent moments as found 
by the beam theory. 

4th One deduction from this (which is also conflrmed by 
extensometer tests) is, that in slabs having equal side and diagonal 
belts of reinforcing rods the greatest actual extensions and true 
stresses in the steel occur at the mid points of those reinforcing rods 
which run directly between the mushroom heads parallel to the 
sides of the panel, and do not occur at the center of the panel where 
mi and m2 have their greatest values. Further, the true stresses in 
the reinforcement are not so large at the edge of the column caps as 
at the points just indicated. Neither of these conclusions is in 
accordance with the beam theory as implied in ordinary formulas 
such as have been frequently adopted in practice in computing slabs. 

5th In making any statement or specification respecting the 
bending moments at any point of a slab, it is essential to state which 
bending moments are contemplated, the true bending moments or 
the apparent moments, with the understanding that the true bend- 
ing moments only are to be used in determining cross sections and 
stresses of steel. Any statement omitting this distinction is ambig- 
uous, and any requirement seeking to proportion cross sections of 
steel to apparent stresses and apparent moments is incorrect. 



poisson's ratio 11 

4. Poisson's ratio K plays an important role in the theory of 
flat slabs and plates, as is evident from equations (1) and (4). Few 
attempts have been made to determine K by directly measuring the 
amount of the lateral contraction accompanying the elongation of 
test specimens, and, were such measurements made, the relative 
dimensions of the cross section of the specimen would need to be 
considered as affecting in a very complicated way the true value of 
K to be derived from observation. Reliable determinations of K 
usually depend upon observations of Young's modulus of elasticity 
E and the shearing modulus of elasticity F. 

It is proven in the general theory of the deformation of isotropic 
elastic solids that all the elastic properties of any such solid are 
determined without excess or defect by its values of E and F, and 
that Poisson's ratio is a function of E and F expressed by the equation 

K + l = \E/F (5) 

There is evidence to show that for concrete K is approximately^ 
0.1*. For steel it is known that K = 0.3 nearly. 

Now it is evident that a horizontal slab of reinforced concrete, 
in which the reinforcement consists of rods, differs from one in which 
its reinforcement is considered to be a simple uniform sheet of metal 
in this, that the former has much less shearing rigidity in resisting 
horizontal forces than the latter, for in it all stresses transmitted 
from one band or belt of rods to any other belt crossing it are trans- 
mitted thru concrete only, as is not the case if the reinforcement 
consists of a continuous sheet. It is evident therefore that the value 
of K which must be employed in applying the foregoing equations to 
reinforced concrete slabs must exceed 0.3, the value required in case 
the reinforcement is a sheet of steel. 

This analysis of the conditions affecting the value of K for a 
reinforced flat slab differs radically from assuming at ramdon that 
because K = 0.3 for steel alone and K = 0.1 possibly, for concrete 
alone, that therefore some intermediate value of K may be correct 
for these two materials combined in a slab. Such an assumption 
is merely a blind guess and has no rational basis. 

As already partly stated, th(^ view h(^re ]nit forth is this: Since 
in any homog(Mi(M)us, isotropic, (^histic material the experimental 
values of E and F perfectly defliie all its (^lastic ])ro]KTties. and since 
we are evidently at liberty to assume our Hat slab as sulIiciiMitly {'luc 
grained in its structure to act nearly like a slab constructed of soinc^ 
sort of h()m()gen(H)us materials, it will \)c ])()ssibl(^ to (le((M-iuint> 

* Turnoaure and Muurcr'H Rcinlorcod Concrete Conatruction 2nd Ed. 1907, p. 210. 



12 poisson's ratio 

certain mean values of E and F which will define its elastic proper- 
ties. It is moreover evident that in a slab, where two kinds of elastic 
solids are combined as they are here, the mean value of F for the 
combination is much more affected by the concrete than is E, which 
latter may be taken as that applying to the steel alone, and, conse- 
quently as unchanged by the combination. It is otherwise, however, 
with F, because the arrangement of the combination is such as to 
require the assumption of a value of F lying somewhere between 
that for steel and that for concrete. Since the latter value is much 
less than the former, the mean value of F is smaller than for steel 
alone. 

This reasoning and other independent theoretical and kinemat- 
ical considerations have led to the same conclusion, viz: that the 
correct value of K for the slab is larger than 0.3. 

Assuming E = 30,000,000, we may compute corresponding 
values of K and F from (5) as follows : — 

If i^ = 0.1 , F = 13,600,000 
If i^ = 0.3 , F = 11,600,000 
If iT = 0.5 , F = 10,000,000 

Were a perfectly complete and accurate mathematical theory 
of the flat slab at our disposal, we might consider every experimental 
test of the deflection of such a slab, and every extensometer measure- 
ment of its reinforcing rods as an experiment for determining the 
numerical value of K, since deflections and extensions would then 
all be known functions of K. Having brought such a rational 
theory to a somewhat satisfactory degree of perfection, the writer 
finds that, in the light of all known tests of slabs, the value that best 
satisfies all conditions is K = 0.5 (6) 

It is possible that this value of the constant K for slabs may need 
some slight modifications hereafter, but for the present this may be 
regarded as substantially correct for mushroom slabs. It may be 
found necessary to assume a somewhat different value for other forms 
of structure, as for example, beam and girder construction. That, 
however, must be determined later. Moreover it must be said that 
this value of K applies to tests made upon slabs from 2 to 4 months 
old, and under loads which have been applied to such relatively soft 
concrete as this for a period of usually not longer than one or two 
days, and of an intensity such as to cause a maximum stress in the 
steel of from 10,000 to 16,000 lbs. per square inch. Less loads on 
better cured concrete, or longer time under load, may show con- 
siderable deviation from this value of K, 



EQUILIBRIUM OF SLAB ELEMENT 13 

How important a factor K is in slab theory is evident on con- 
sidering equations (4) which show that in a square panel uniformly 
loaded the true moments as shown by the elongations of the rein- 
forcing rods at the center of the panel and over the centers of the 
columns are only one half the corresponding apparent moments 
derived from considering the moments required to hold the applied 
forces in equilibrium, this being on the assumption of course that 
K = 0.5. 

5. In order to derive the general differential equation of shears 
and moments in any rectangular panel in an extended horizontal 
plate or slab, take the axes of x and y in the neutral plane of the 
plate and parallel respectively to the longer and shorter sides of the 
panel with the origin at its center before flexure occurs, and assume 
that they remain fixed with reference to the points of support of the 
panel. Then during flexure the center of the panel and all other 
points of the slab or plate not in contact with the fixed points of 
support will attain some deflection z, of amount to be determined 
later. Take z positive downwards. 

Then dxdy is the horizontal area of an element of the slab 
bounded by vertical planes, and if d be the effective thickness of the 
slab or plate, the areas of the sides of this element which are respec- 
tively perpendicular to x and y are ddy and d8x, while ddxdy is the 
volume of the element. 

We proceed to obtain the equations of equilibrium of this ele- 
ment of the slab as follows: — 

Let Si and S2 be the total vertical shearing stresses per unit of 
width of slab for sections perpendicular to x and y respectivel3\ In 
case these shears are variable, as they are in a continuously loaded 
slab, they respectively contribute elementary forces tending to move 
the element vertically, of the following amounts : 

8 Si 8 8-2 

8y8x, and 8x8y 



8 X 8 y 

Assume that the slab carries a uniforml}^ distributed load of q pounds 
per square unit of area. Then the load upon the elementary iwcii 
8x8y is q8x8y, and the equation of equilibrium of tlu^ vertical forcn^s 
acting on tlu^ element reduc(^s to this: 

6 Si 8 So 
+ --^^ + '/ = (7) 

ox !l 



14 EQUILIBRIUM OF SLAB ELEMENT 

in which Si and S2 are taken as positive when they are such as would 
be produced in the slab bj^ the loading q in case it were supported at 
the origin only. 

Let nil and m2 be the apparent moments per unit of \\ddth of 
slab of the applied forces which tend to bend those lines in the slab 
which before bending are parallel to x and y respectiveh\ Take 
them as positive when they tend to make those lines respectively 
concave upw^ards. These are the moments obtained by multiphdng 
the total applied tension per unit of ^ddth of slab by the vertical 
distance jd from the center of the reinforcement of the slab to the 
center of compression in the concrete as given in (2) . These moments 
are not identical in a slab with the true resisting moments rrii and m2 
in the same directions, which latter are the moments obtained by 
multiph^ing jd by the actual tension in the steel per imit of width of 
slab, which last is to be correctly computed bj^ taking the product 
of the area of steel per unit of ^^idth and its elongation multiplied 
by E its modulus of elasticity' as shoTvn in (3). 

Again, let n be the tT\isting moment per unit of T^'idth of ver- 
tical section of slab cut bj^ planes perpendicular to either x or y, and 
acting about either x or y, which moment n is regarded as due to the 
variation of the vertical shearing stress Si when y varies, and to the 
variation of S2 when x varies. The moment n is held in equilibrium 
by horizontal shearing stresses in these same sections, which are 
opposite in sign above and below the neutral surface. Let t be the 
total horizontal shearing stress per unit of T^ddth of slab in the rein- 
forcement on one side of the neutral plane, then: 

n = i A j d (8) 

At any point xy this horizontal shearing stress t must be the same 
for the section perpendicular to x, as for the section perpendicular 
to y, because in every state of stress the tangential components are 
equal and of opposite sign on any two planes mutually at right 
angles. Consequently the moment n is the same about x as about 
y, as has been assumed in (8). 

It is implicitly assiuned in (2) and (3) that the concrete on the 
same side of the neutral plane as the reinforcement is ineffective 
and that its resistance is negligible, so that on that side the resistance 
of the reinforcement alone counts. This condition actuallj^ occurs 
only after a state of quite considerable stress obtains, and of itself 
affords a sufficient reason why the formulas based on it fail of accu- 
rately representing deflections and elongations at small loads and 
low stress. 



DIFFEHENTIAL EQUATION OF MOMENTS 15 

The elementary couples acting on the vertical faces of the 
element which are in equilibrium with those arising from the shear- 
ing stresses are : — 

(5 mi 5 n \ 
-f I dxdy about y, and 
d X d y / 

(5 m2 5 n \ 
-f I dx8y about x; 
by b x / 

while those arising from the shears themselves are : — 

Si bx by and S2 bx by. 

Consequently the equations of equilibrium of the couples acting on 
the element reduce to the following: 



+ -^ + si = I 

i (9) 

+ + S2 = I 

by b X 

Differentiate equations (9) with respect to x and y respectively 
and substitute in (7), and we obtain 

b^ nil 5^ n b^ m2 

+ 2——- + -T-^ = q (10) 



5 


mi 


b 


X 


b 


m2 



b 


n 


b 


y 


b 


n 



b x^ bx by b y^ 

which is a general differential equation of the apparent moments 
of the appHed forces which exist in a uniformly loaded slab in 
terms of rectangular coordinates. From it the differential equa- 
tion of the deflections may be derived as follows: — 

6. To obtain the general differential equation of tlio deflec- 
tions of a slab, note that from geometrical considerations such as 
are familiar in the theory of beams we have 

Ri ei = i (I = Rofo (11) 

in which h\ and R2 ^i^'^- H^^- nidii of curvatur(» of scH'tions of \ho 
neutral surface by vertical ])lanes liaraflel to .r and // res])(H'ti\(^ly; 
and id is the distance from the center of iUv r(>inf()rc(MiuMi( to ihc 



16 MOMENTS AND CURVATURES IN SLAB 

neutral surface. In equations (la) replace pi and p2 by values 
^iven in (2), and ei and 62 by values taken from (11) and we have: — 

(1 - K')m, = EA ijd^ ( 1- + — ) 

\Ri R2/ \ 

^ (12) 



2\_ TTT ^ ••72 



(1 — K^)m2 = E A ijd 



But from the theory of curvature 



\R2 R\/ 



1 ^ h^ z 1 ^ h^ z 

— = Z , and — = 1_ -— (13) 

R\ d X2 R2 d y 



Also write for brevity I = A i j d (14) 

■Then we have from (12), (13) and (14): 

8' z d' z \] 



- / 8' z 8' z \ 
(1 — K^)m, = Z.EI I + K I 

\8x' ^ 8y' ) 

, / 8^ z 8^ z \ 

(1 — K^)m2 = ±EI[ + K I 

\8 y' 8 x' / 



(15) 



By the fundamental equations of elasticity we also have 

(8u 8v \ 
— + — ) (16) 
oy ox / 

in which F is the shearing modulus, 63 is the horizontal shearing 
deformation of the reinforcement for two vertical planes one unit 
.apart horizontally, and 

_\ 8z \ 8z 

u = _i d — , V = __i d — (17) 

8x 8y 

.are the deformations along x and y respectively, due to the vertical 
-distance id of the reinforcement from the neutral surface. 



From (16) by help of (17) we have 

8x8y 



t = Ji2Fid (18) 



DIFFERENTIAL EQUATION OF DEFLECTIONS 17 

In (18) replace F by its value obtained from (5), and then sub- 
stitute the resulting value of t in (8) : — 
we then have 

EI d^ z 

n = • — (19) 

1 + K 8xdy 

From (15) and (19) obtain values of the second differential 
coefficients of the moments appearing in (10), which on being intro- 
duced into (10), transform that equation into the required general 
differential equation of deflections as follows: — 

8^ z b^ z b^ z (1 — i^') 

h 2 \ = q (20) 

b x^ bx%^ b y"" EI 

which is a partial differential equation of the fourth order that must 
be satisfied by the coordinates x y z oi the neutral surface of any 
uniform plate or slab initially flat, when deflected by the applica- 
tion of a uniformly distributed load of intensity q, and supported in 
any manner whatever. 

It may be shown that any deviations from strict accuracy by 
reason of local stretching of the neutral surface (here neglected) are 
small compared with corresponding deviations in beam theory. 

7. The solution of the general differential equation of deflec- 
tions (20) for the case of a horizontal slab carrying a uniformly 
distributed load and supported on rows of columns placed in rec- 
tangular array and having the points of support all on the same 
level, will now be considered. 

The integration or solution of (20) would, since it is a partial 
differential equation, introduce arbitrary functions of the independent 
variables x and y whose forms would need to be so determined as to 
cause the solution to satisfy the conditions imposed by the i)()sitiou 
and character of the supports at certain points, or along certain 
lines. It would be possible to expand these functions in terms of 
ascending whole powers and products of x and y, and, in case tlu^ 
supports are symmetrically situated with respect to the axes, (h(^ 
expansions will contain no odd powers of x or //, IxH'ausi^ {\w vnhie 
of z must remain unchanged by changes of sign of (m(1um- x or //. or 
both X and y. Any form of polynomial expansion ^^llil'h satisfies 
(20), and also all the conditions of any givcMi case, must \)v the correct 
solution for that cas(% for, the solution of any givcMi case nuist be 
unique. 



18 GENERAL EQUATION OF DEFLECTIONS 

Instead therefore of carrying thru the tedious analytical devel- 
opment involved in solving (20) mathematically and then applying 
it to the case we are treating, we shall at once write down the form 
of solution that applies to the case in hand and verify the fact that 
it satisfies (2) and all the required geometrical conditions. It will 
therefore be the solution sought for, which might also have been 
obtained by the somewhat intricate analytical processes involved 
in the intregation of such differential equations as (20). 

Assuming at first that the slab is unlimited in extent and uni- 
form thruout in the distribution of its reinforcement and loading, 
and that the parallel rows of supporting columns divide the slab 
into equal rectangular panels, we shall find a solution in which every 
panel is deformed precisely in the same manner as every other. 
Modifications made later will render it possible to take account of 
variations and irregularities in the distribution and arrangement of 
the reinforcement, and to estimate to some extent at least the effect 
of loading only one or more panels. 

Let 2a be the length and 25 be the breadth of a panel; then the 
equation of its neutral surface, referred to axes parallel to its sides 
and to an origin fixed in space at the center of the neutral surface of 
the panel before deflection, is: — 

48 EIz = g(l — K^) \{a^ — x^f + (6' — y^] (21) 

This is the correct solution of (20) not only because it satisfies 
(20), as it will be found to do by trial, (and just as many other func- 
tions of X and y do also) but it also satisfies all the other conditions 
required by the case proposed, viz. : 

1st z = when both x = ^a and y = + b: 
because there must be no deflection at these points of support which 
are on the same level as the origin. 

2nd dz/ dx = 0, when x = 0, and also when x = ^ a; as well as 
dzydy = 0, when y = 0, and also when y = dl 6; because straight 
lines drawn in space to touch the slab across its edges, and across 
its mid sections parallel to those edges, must all be horizontal by 
reason of the symmetry of the slab on each side of its edges and mid 
sections. That these conditions hold is evident from the following 
equations derived from (20) : 

q X [x — a ) 



5x 12 E I 

d z (1—K^) 
dy 12 EI 



(22) 



qy iy^ — h^) 



TRUE STRESSES AND MOMENTS. LINES OF CONTRA-FLEXURE 



19 



It is of interest to note that the sections of this surface made 
by all vertical planes parallel to the axes of y, i. e., by a; = constant, 
are precisely the same except in position, since their equations differ 
by a constant only. The same is true of sections parallel to x. It 
thus appears, that, in a square panel where a = b, the surface may 
be regarded as a ruled surface described by using the two of these 
curves on a pair of parallel sides of the panel as directrices and a 
third one of these curves as a ruler sliding on the first two in such a 
manner as to remain parallel to the other pair of parallel sides. 

The deflections at the center of the panel and middles of the 
sides are: 

Xtx = = y, 4&E I z = q {l-K^) {a" + h^) 

ktx = ±a,y ^ 0, ^SE I z = q (1-K^) 6* 

Atx = 0,y = ±h, ^SE I z = q (1-K^) a^ 

so that in a square panel the center deflection is twice the mid edge 
deflection. 

Differentiating equations (22) we have by help of (11), (13), 
(14) and (3) : 

(l—K^) 

/o 2 2\ 

q (3x — a ) 



ei 



id 

id 



+ id 



d' z 



Co = — = '^ i d 
R 



8x' 
d' z 



) 
^2 


5x' 


nil 


= q [Sx 



12 



12E Aj d 

12 E Aj d 
-a') 



( 



■ (23) 



q (3/ - h') 



(23a) 



rrio = 



12 



qiSy'-b') 



in which the ambiguous signs are to be so taken that //?i and ))i-2 in 
(15) will be positive at a; = = ?y, and negative at .r = dl a and 

y = +b. 

From (23) it appears that extcMisions vanish and coiitra-tU^xui'i* 
occurs at lines lying in v(*rti(;al ])lan(\s whose (M|ua.tioiis luv 

X = + I aV'^ and y = -^ I hV:] (24) 

it thus appears thai llie shih is sulxUvidiul by tliesi^ Hues (24) 
drawn ])arallel to the edges into a i)atl(^ni which consists of a riH*t- 
angle ()ccu])ying the middle i)art of eacli i)aiiel, of a siz(» I (})/;\ by 



20 APPARENT MOMENTS AND SHEARS 

f hVs, i- e., of the same relative dimensions as the panel itself, and 
bounded by lines (24), which rectangle is concave upward thruout. 

On all four sides of this central rectangle are rectangles of saddle 
shaped curvature directly between the central rectangles of adjoin- 
ing panels, while each point of support is situated in a rectangle 
which is convex upward over its entire area, of dimensions 
2a(l— i V^3)by26(l— i ^3). 

From (22) we obtain the equation 

5^ z / dx8y = (25), 

hence b}" (18) and (19) it follows that 

t = = n, (26), 

from which it appears that there is no horizontal shear in the steel, 
and no twisting moment in vertical planes perpendicular to x or y. 
This would be otherwise evident from considerations of sj^mmetry. 
It will be shown that this is not true of all other vertical planes. 



(27) 



Again from (15) and (23) we have 

m, = i,q [3x^ — a^ + K{^^ — h") ] 

m2 = h q [K{Sx^ — a') + 3/ — 5' ] 

in which we have omitted the sign +. as superfluous. 

From (9) bj^ help of (26) and (27), we have 

5mi 5m2 

— Si = = \ qx, and — S2 =- = \ qy (28) 

hx by 

from which it appears that any strip of the panel parallel to x or y^ 
and one unit wide exerts a shear at its ends such as it would if it were 
an isolated beam loaded uniformly with an intensity of ^q per unit 
of length. According to this, a total shear of g a 6, which is one 
fourth of the total load carried by the panel, appears at each edge of 
the panel, this total shear on each edge being uniformly distributed 
along it. 

It is seen therefore that the form of solution which w^e are 
investigating implicitly assumes that at each edge of the panel there 
is some auxiliar}" form of structure that will bear the shears coming 
to it from each side and at the same time assume the curvatures and 
deflections contemplated in (21). This will immediately engage 
our further attention. 



SIDE BELTS 21 

8. In order to investigate more fully the deflections, stresses 
and strains in the side belts of any panel directly between the mush- 
room heads, let us consider the results just reached somewhat more 
fully. 

The conclusion drawn from (28) was, that a panel with rein- 
forcement distributed with perfect uniformity thruout would require 
to be supported by a narrow auxiliary girder extending from column 
to column along each side, and of such resisting moment as to take 
on, under its load, the precise curvature required by the neutral 
surface in (21), which curvature must be produced by a uniformly 
distributed load of 2 q ah, one half of it coming from each of the two 
panels beside it. 

It seems then, that up to this point, we have in reality been 
treating the theory of the continuous uniform slab with specially 
designed continuous beams supporting its edges, without as yet 
investigating those beams in detail. But since no such beams in 
fact exist under the flat slab, it is clear that the side belts of the slab 
lying directly between the extended heads of the columns must 
discharge the functions which would be discharged by the auxiliary 
beams just spoken of. Such functions must necessarily be added 
to those already discharged by those belts in supporting the loading 
which rests directly upon them. In order that this may occur in a 
manner readily amenable to analysis, the extended stiffened head- 
ings of the columns which constitute the mushrooms should in 
general be approximately of the diameter required to support the 
ends of a belt of reinforcing rods forming a flat beam which fills the 
width along the edge of two adjacent panels between the two lines 
of contra-flexure on each side of that edge, as given in (24). 

This requires that the mushroom head should have a width of 
at least (1 — | Vs) = .423 of the width of the slab between col- 
umns. For reasons that will appear later, it is current practice to 
make these heads not less than I'o = .437 of this width. 

The lines of contra-flexure in (24) have a fixity of position, (in a 
flat slab constructed with mushroom heads of this size and stift'ness,) 
under single panel loads, that does not (^xist in a. uniform slal), or 
where the headings are not so stiff. It may b(^ n^adily shown l)y 
Mohr's theorem resj)ecting (U^HtH'tion curves as stM'ond monuMit 
polygons, that where there are lai'ge sudden ('hang(\s in th(> nia.iiiii- 
tude of the monuMit of inertia /, sucli as (»xist in this caso at {\\v liiu^s 
of contra-flexure at tluMulgc^s of th(* mushroom. \\\v hn(>s of contra- 
flexure nnnain fixed. Hut. in systeins wIumc the diaini'tcr o^ \\w hvixd 
is smaller tlian "iven al)()ve, or its stiffness is nnich icMhici'd. thesi^ 



22 SIDE BELTS 

lines may be removed to greater distances from the center in loaded 
panels surrounded by those not loaded than when all are loaded, 
thereby increasing the deflections and stresses in a single loaded 
panel over that of a uniformly loaded slab of many panels. 

The lines of contra-flexure in (24) separate the slab into areas 
which are largely independent of each other, since no bending 
moments are propogated from one to another. The only forces 
crossing these lines of section are the total vertical and horizontal 
shearing stresses. The horizontal shears (which are unimportant so 
far as deflections go) will be considered later so far as may be necessary, 
but the vertical shears found by (28) are of prime importance. Let 
us then consider one of these side belts. 

In any extended slab with its panels all loaded uniformly 
thruout, the vertical shear must vanish at all points along sections 
made by vertical planes thru the centers of columns at each side of 
any panel, as appears by reason of symmetry of loads. Let the 
edges of the side belts be situated at some given distances, say Xi and 
yi on each side of the centers of all the panels, where Xi and yi are 
not necessarily the values of x and y given in (24), altho those 
values are also included in this supposition. Then by (28) there is 
a uniformly distributed vertical shear of intensity i q yi along the 
edge of the belt at y = yi, even tho the reinforcement in the side 
belt may be greater than that in the central rectangle, for the devia- 
tions caused by the irregularity of its distribution may be regarded 
as unimportant and practicably negligible. 

It may then be assumed that any side belt parallel to x must 
carry, in addition to that already provided for in (21), a total loading 
of q yi per unit of length, uniformly distributed along the two edges 
that are parallel to x. Now since the width of this belt is 2(6 — yi), 
the load already provided for in (21) is ^q per unit of area, or q(h-yi) 
per unit of length parallel to x, which added to that arising from the 
shears just mentioned makes a sum total of g 6 per unit of length of 
belt, which it will be noticed is independent of the width of the belt. 
In other words, any such belt must support a load of one fourth of 
the total load on the two panels of which it forms a part, or one half 
of all that lies between the panel center lines which are parallel to 
it on either side. This in effect transfers the entire loading of the 
slab to the side belts by the agency of the shearing stresses. It does 
this in such a way that one half of the total loading of the entire slab 
is carried by one set of side belts, and the other half by a second set 
which crosses the first at right angles. 



DEFLECTION OF SIDE BELTS 23 

In those parts of the slab area where these sets of belts cross, 
forming the heading of the columns, the loading is superposed also. 

The preceeding investigation of the shears at the edges of side 
belts and their loading is independent of their width and of the posi- 
tion of the lines of contra-flexure, but their width will be assumed in 
what follows to be determined by the position of those lines as shown 
in (24) on account of the independence of action of belts of their 
width, as previously explained, where it was shown that no bending 
moments are propogated across those lines. 

The question now arises, how the vertical shears at the edges 
of the side belts are distributed across their width and carried by 
them. Since by symmetry of loading, etc., there is no vertical 
shear at the edge of the panel where y = h, the shear must diminish 
from each edge of a belt to zero at that line. If it be assumed to 
diminish uniformly, that is equivalent in its action to a uniformly 
distributed load on the belt, which may be assumed in computation 
to replace the shears at the edges. Whether it will be so distributed 
or not depends upon the stiffness of the mushroom head and the 
«mallness of its flexure. Extensometer measurements on the rods 
of the side belt of the floor slab of the St. Paul Bread Company 
Building by Prof. Wm. H. Kavanaugh show beyond question that 
in the mushroom system the load is so distributed. Other exten- 
someter measurements to which the writer has access also show that 
in systems in which the heading of the column is not so stiff as this 
the distribution of loading cannot be taken as uniform over the side 
belts. 

Now the belt parallel to x was shown to carry a load per unit of 
length oi qh and to have a width 2(6 — ?/i), in general, or a width 
26(1 — \^Z) for the belt between the lines of contraflexure; hence 
the intensity of the loading on this belt is q 6/2(6 — ?/i), instead of 
g, as it would be in a uniformly loaded panel duly supported at its 
•edges by beams from column to column. Let 2 A, designate the 
area of the effective right cross section of the steel in the entire width 
of a side })elt regarded as forming a single sheet of metal of the width 
of the belt; then 2^/2(6 — yi) is the effective right cross section 
per unit of width of l)elt, and we may Avrite (14) in the form 

/ = ?: i r/' 2/1 /2(6 — 7/0 (29) 

We sliall considei- admissible \'aiiies of 2.1 IntcM'. 

Since the deflection of the side bells \\\[\\ bo lakcMi iiulei)oiul(Mit ly 
of th(> r(>st of the slab, 1(M, (li()S(> \'alu(\s for (li(> iiihMisity o{ loading 
xmd tiie inoinoni of iiioi-lia, (29) be iiit I'odiicod into ('JH. 



24 STRESSES IX SIDE BELTS 

We then obtain an expression for the law governing the deflec- 
tion of that part of the side belts parallel to x which lies between the 
mushroom heads, and is bounded by hues of contra-flexure, \iz : 

(1 - K-) q h 



48 r i j d' ^A 



[(a- — ^-)- + Qr — irr] (30) 



with a corresponding equation for the side belts parallel to y. which 
may be obtained by replacing q h in 30' by q a. Call this second 
eciuation (31\ Xow '30^ and 31' would hold thruout the entire 
length of these belts from colunni to column were they entirely 
separate from each other and from the diagonal belts where they 
cross each other. It T\-ill be necessary later to obtam the equation 
which holds true where these belts cross and combine with each other. 

9. Practical formulas for the stresses in the steel and concrete 
of side belts between the lines of contra-flexmT will now be obtained 
from '30) and ''31). 

In order to do this, consider the summation in (30) expressing 
the effective cross section of the steel in the mid area of the side belt 
regarded as forming a single uniform sheet, that mid area being 
bounded on all sides by lines of contra-flexure. 

It is to be noticed that the factor (1 — K~) of (30) takes into 
account the fact that the lattice of rods forming the reinforcement is 
less effective than the same amount of metal in the form of a sheet, 
the only question left being this: Will the great irregularity of 
distribution of the reinforcement in this area cause it to act differ- 
ently to any noticeable extent from the manner in which the same 
amount of metal would act were it possible to distribute it uniformly 
over the entire area? There are strong reasons which go to sustain 
the ^iew that this irregularity of distribution is negligible in the 
standard mushroom slab, at least for loads less than those that 
stress the steel below the ^ield point, or do not stress the concrete 
for too long a time while it is imperfectly cured. On examining a 
diagram of the reinforcing rods of a slab made with square panels 
of such proportions that the width of the belts is one half the cUstance 
between columns, then the pattern previously mentioned into which 
it would be di^aded by these belts \\ill be seen to consist of equal 
scjuares whose edges are equal to the ^idth of the belts, with one 
central scjuare in each panel concave upwards, and one half of each 
of the saddle shaped squares which border it. also lying within the 
same panel, and one quarter of each of the four convex squares at the 



MEAN REINFORCEMENT OF SIDE BELT 25 

head of each of the columns at the corners of the panel, also lying 
within the same panel, see Fig. 3, page 7. 

Each side square will be found in this case to have double (or 
two belt) reinforcement over one half or its area, single belt rein- 
forcement over a diamond occupying one fourth of its mid area, and 
triple reinforcement over four triangular areas along its sides which 
together cover one fourth of the square. This gives a mean value 
of 2 A = 2 yli in which ^li is the total right cross section of the rods 
in the side belt. 

The belts in the standard mushroom are, however, not so wide 
as this, since that system simply requires that the edges of the side 
and diagonal belts intersect in a single point. Fig. 2, instead of forming 
four areas of triple reinforcement on the sides. This makes the 
width of the singly reinforced diamond sufficient to just reach across 
the side belt. In this practical case we find that very approximately 

2^ = 1.5 Ai (32) 

in which, as before, Ai is the total right cross section of the side belt 
in square inches. It is evidently impossible for this single side belt 
of rods which crosses the diamond, to elongate without a correspond- 
ing equal elongation of the double reinforcement on all its sides, or 
at least it is impossible for readjustments to take place in an}- short 
time such as will make these direct deformations within the diamond 
larger than those in the areas along side of it, or before somewhat 
more permanent deformations have taken place in the concrete. 

In cases where the column heads are smaller than the standard, 
and the side belts still narrower, not only may XA become much 
less than 1.5 Ai but the belt become so weakened near the central 
diamond as to render it very questionable whether the irregularity 
of distribution of steel in the area considered may be safely disre- 
garded. Diminution of the size of the heading thus not only dim- 
inishes cantilever action, but reduces the effective resistance of the 
reinforcing steel. Not much diminution of the size of head would 
be required to nnluce the value of SA to an amount as small as .li. 

Introducing the estimate given in (32) for the standard mush- 
room into (30) we derive by (23), (23a) and (3), for that ])art of the 
side belt parallel to x between x = + 5 aV^3 and x = — J a>/3, 

5-.- (\-K'')qb ., 

/^ = E e, = + E i (I ., = ± (3.r- a') 

b x" IS 7 (I A I 



Ml = ]..") A , i (/ f. = + (I h C-lr (D 

12 



(33) 



26 TRUE STRESSES IN SIDE BELT 

in which Mi is the total true moment of resistance of the side belt, 
fa is the true stress per square unit of the reinforcement in the side 
belt, and 1.5 Ai is the effective right cross section of the reinforce- 
ment. This is independent of y as before noted, showing that the 
values of /s and Si are the same for one rod as for another, but they 
attain their greatest values at the mid length where x = 0. If units 
be pounds and inches, and we assume j = 0.91 for the very small 
percentage of reinforcement of the standard mushroom system, then 
by (33) and (6) the practical formulas for design are: 



Sqa^h W L 

/s = 



4 X 18 X 0.91 di Ai 175 di Ai 

W L 

Ml = l.bAijdif, = 



128 



(34) 



in which /g is the true stress in the steel, and Mi is the true bending 
moment of the effective cross section 1.5 Ai of the steel in the entire 
belt as shown by the elongation (at mid span) of the rods in a side 
belt of length L, where L is either 2a or 26, and W = 4: q a h is the 
total load on the panel in pounds, where di is the vertical distance 
from the center of the rods in the single belt at mid span to the top 
surface of the slab. 

While the values obtained from (34) are conservative for j = 0.91, 
corresponding to a percentage of reinforcement for one belt of less 
than 0.25%, (34) should be regarded merely in the light of a speci- 
men equation for that percentage, and any slab where the percent- 
age differs materially from that assumed value should be submitted 
to separate computation in the same manner. 

Values of j are given for beams by Turnearue and Maurer in 
their '^ Reinforced Concrete Construction," page 57, for different 
percentages of reinforcement on the straight line theory, which 
latter is now accepted usage. As already stated, standard mush- 
room design makes the percentage of reinforcement for warehouse 
floors where the panels are, say 20' x 20', as low as 0.25% or less, at 
the middle of the side belts, reckoned on the beam theory. But in 
heavier and larger construction it may reach 0.33%. 

We have taken the mean available steel in the belt as 1.5 A i, 
hence the mean slab reinforcement will not be less than 1.5 x 0.23 = 
0.4% in the side belt areas between lines of contra-flexure. 

In case we assume the ratio of E^ for steel to E^ for concrete to 
be 15, as is often prescribed, we find the above stated value of j as a 



THEORETICAL AND EXPERIMENTAL STRESSES COMPARED 27 

good mean value, which will be less in cases where the percentage 
of steel is greater. The small percentage of steel and great relative 
thickness of concrete is one of the distinguishing features of the 
standard mushroom design. 

We may write (34) in the form: 

W L W L 

/s - ,and M[ =A,jdf, = .... (34a) 

175 di Ai 192 

in which Mi is the true bending moment of the actual cross section 
^1 at mid belt. We have written this modification of (34), not for 
use in design, but merely for the purpose of instituting a comparison 
with empirical formulas obtained by Mr. Turner to express the 
results of numerous tests made by him. On pages 26 and 28 of his 
'^Concrete Steel Construction" he has given equations expressing 
the values of stresses and moments in mushroom slabs which in our 
notation may be written as follows: — 

WL W L W L 

Mi = Aij df,= , and /, = = . (35) 

200 200x0.85^^1 170 dA^ 

in which he has assumed 0.85 as a mean value of j. 

It is seen that equations (34a), obtained from rational theory 
alone, are in practical agreement with (35), which were deduced 
from experimental tests of mushroom slabs, where the numerical 
coefficient introduced is entirely empirical. 

As will be seen later, (34) is the equation which ultimate^ con- 
trols the design of the slab reinforcement; so that the agreement of 
these two entirely independent methods of establishing this funda- 
mental equation cannot but be regarded with great satisfaction as 
affording a secure basis for designs that may be safely guaranteed 
by the constructor, as has been the custom in constructing standard 
mushroom slabs. 

The slab theory here put forth diverges so radically from the 
results of beam theory that we introduce here the following compar- 
ative computation of th(^ smallest values of true IxMuling moment 
and stress in steel, which can be obtained by l)(^aiH (iHH)ry for \\\v 
side belt parallel to x, as follows: — 

That i)art of the side belt Ix^Iwimmi (he lines of ('()ii(i-;i-ll(>\iiri> is 
simply siii)])()rt(>(l at its ends by shearing sti-esses, and so nia>" be 
taken to be a sim])le beam resting on sui)i)()i-ls at thes(> end lint^s. 



28 STRESSES BY BEAM THEORY AXD BY TEST 

Hence the true stress /^ and the true bending moment M at the 
middle of this simple uniformly loaded beam may be computed from 
the equation, 

M' = A.jdf, = I TT^'Z' (36) 

in which M is the total moment of resistance. 

Ai is the total right cross section of the reinforcement. TT is the 
total uniformly distributed load, and L is the length of the beam. 
The length of the simple beam in that case is evidently the distance 
along X between lines of contra-flexure. viz. L =faV^3=3l' V^3. 
where L is the edge of the panel, and the total load at most will be 
that already proven to be carried by the side belt viz. q h per unit of 
length, or a total for a span L oi W = qh L = f 5 a 6 ^3 = f TT V^3 
where TT = 4 qah is the total load on the panel, hence 

W L 

M' = A.jdu = (36a) 

48 

It thus appears that according to simple beam theory the true 
stress, or the cross section of steel required in the belt, is four times 
that obtained by slab theory as shown by (34a). Since (34a) is m 
good accord with experimental tests, this comparison justifies the 
statements made near the beginning of this paper respectmg the 
inapplicability of beam theory to the computation of slab design. 

The floor of the St. Paul Bread Co. Building, previously men- 
tionedjis a rough slab 6'' thick, and has panels 16' x 15', ■v\'ith ten 
3/8'' round rod reinforcement m each belt, built for a design load of 100 
pounds per square foot: constructed in winter and frozen, the final 
test was not made until the end of its first summer after unusually 
complete curing, such as might make the value of K given in ('6) 
not entirely applicable. In one long side belt, extensometer measure- 
ments were made at the mid span on three rods, (1) a middle rod, 
(2) an intermediate rod and (3) an outside rod of the belt, ^ith the 
following results for the given live load in pounds per square foot: 



Live Loads 


108.4= 


316.8= 


416.8= 


/= = Eei (1) 
(2) 
(3) 


7650 
7080 
7320 


15000 
14190 
13920 


17940 
16470 
17160 


Average 


7350 


14370 


17200 


/s by (34) 


5000 


14440 


19000 



STRESSES AT YIELD POINT 29 

The observed results are seen to be in excellent agreement with 
those computed from (34) for the heavier loads, while any disagree- 
ment is on the safe side. Agreement is not expected for light loads. 

The accuracy and applicability of (34) and preceeding formulas 
is dependent on the fixity of the lines of contra-fiexure (24) which 
were previously stated to be practically immovable because of the 
sudden large change of the moment of resistance of the slab at those 
lines. That fact may be put in a more definite and convincing 
form than has been done so far. Consider for a moment that form 
of continuous cantilever bridge where there are joints between the 
cantilevers over the successive piers (which are in the form of a letter 
T) and the intermediate short spans which connect the extremities 
of the cantilevers. At such joints the resisting moments vanish, and 
they form in a sense artificially fixed points of contra-flexure. The 
same thing approximately occurs at the edge of the mushroom, 
because there the reinforcing steel rapidly dips down from a level 
above the neutral plane to one below it, and the sign of the moment 
of resistance changes thru zero at that edge. 

Furthermore, it may be proper to state in this connection that 
the foregoing theory has been developed in consonance with the 
general principles of elasticity, and that somewhat different condi- 
tions and relations are thought to exist when the steel at the middle 
of the side belts reaches its yield point, as it does in advance of the 
rest of the reinforcement. As the yield point is reached equations 
(34) no longer hold; for, as will be seen more clearly later, the single 
belt of reinforcing steel, which crosses the circumference of an ap- 
proximately circular area of radius L/2 about the center of each 
column, will everywhere reach the yield point at practically the same 
instant, and if loaded much beyond this will develop a continuous 
line of weakness there. The equations that hold in this case will be 
approximately those due to the actual cross section A i of the belt, in 
place of (34), which contain the effective cross section, viz: 



Zqa'b W L 

/s = 



4x 12x0.91 r/, Ai 117^/, ^i 

W L 

Ml = A,j(l,f, = - - 

128 



(37) 



whi(;]i may be regarded as (»x])ressiiig tlie relations that i^xist at (lie 
limit of the (elastic strength of \\\v slal) aiul tlu^ b(\ii;imiing of ])i»rnia- 
nent deformation, tho not necessarily of ('()lla])Si». 



30 STRESSES IN CONCRETE 

The percentage of reinforcement in standard mushroom slabs 
is small enough to make their elastic properties depend upon 
the resistance of the steel. The stresses in the concrete may then be 
be computed from those in the steel, but many uncertainties attend 
any such computation. It is usage, fixed by the ordinances of the 
building codes of most cities to require the application of the so 
called '^straight line theory" in such computations, not because that 
will give results which will be verified by extensometer tests of com- 
pressions in the concrete, for it will not, but because it is definite 
and on the side of safety. Furthermore it is usually prescribed 
that the ratio of the modulus of elasticity of steel divided by that 
of concrete shall be assumed to be 15, where the moduli are unknown 
by actual test of the materials. This is usually far from a correct 
value. The consequence is that the results of computation of the 
stresses in concrete are highly artificial in character, and should not 
be expected to be in agreement with extensometer tests. With this 
understanding the computed stress in the concrete at the middle of 
the side belt will be found as follows : — 

Let id be the distance from the center of the steel to the neutral 
plane. (It happens to be more convenient in this investigation to 
use this distance id here and in our previous formulas than to intro- 
duce the distance from the neutral axis of the slab to the compressed 
surface of the concrete, as is done by many writers, under the desig- 
nation k d. These quantities are so related that i -\- k = 1 ). 

Then, as is well known from the geometry of the flexure of 
reinforced concrete beams, in case tension of concrete is disregarded, 

k E, 

/c = -'~fs (38) 

i E, 

where the subscripts c and s refer to concrete and to steel 
respectively. 

Applying (38) to the greatest computed stress /s = 19000 in 
the St. Paul Bread Go's Building, gives a computed stress /c = 492; 
but taking the greatest observed stress /s = 17940 gives /c = 465 lbs. 
per sq. inch, as the greatest computed compressive stress in the 
concrete at the middle of the side belt, if i = 0.72. 

The tensile stress across the middle of the side belt at the 
extreme fiber of its upper surface is fixed by the curvature of the 
vertical sections of the slab in planes that cut the side belt at right 
angles. As stated previously all such planes make cross sections of 
the side belt that are identical in shape. That is a consequence of 



DEFLECTIONS IN COLUMN HEAD AREAS 31 

the conclusion reached previously, that all the rods in the side belt 
are subjected to equal tensions. The curvature of these sections 
is controlled by the stiffness of the mushroom heads, which is so 
great as to make the curvature very small. No considerable tensile 
cross stresses are consequently to be apprehended; but in case the 
stiffness of the head were to be decreased, stresses might arise such 
as to develop longitudinal cracks over the middle rod of the side belts. 

10. In order to obtain practical formulas for the deflections and 
stresses in the steel thruout the areas at and near the tops of the 
columns where all the belts cross each other, and lying between lines 
of contra-flexure, we shall have recourse to (30) and (31) which are 
here superimposed on each other, and combined together. Were 
there no steel here in addition to the side belts, that superposition 
could be correctly effected by writing a value of z whose numerator 
would be the sum of the numerators of (30) and (31), for that would 
superpose the loads of the two side belts, and thus place the total 
required loading upon this area as previously explained; and then by 
writing for a denominator the sum of the denominators of (30) and 
(31), for that would superpose and combine the resistance of all the 
steel in both belts. But such a result would leave out of account the 
reinforcement arising from the diagonal rods, and the radial and 
ring rods, which should also be reckoned in as furnishing part of the 
resistance. 

Supposing this additional steel to be distributed in this area in 
the same manner as is that of the side belts, a supposition which is 
very close to the fact, we may write 

(1 — X') ^ (a + 6) 

48 E ijcfX A 

in which 2A is the cross section of the total reinforcement in this 
area regarded as forming a uniform sheet, i and j stand for mean 
values that have to be determined by the percentage of reinforce- 
ment and its position, while d is the mean distance of the centcM- of 
action of the steel al)ove the lower compressed surface of the con- 
crete at the point xy. 

Wv, may conservatively assume in the standard mushroom 
that the center of action of the steel is at the center of {\\v third \i\ycv 
of rods from the toj), as will ap])ear mon^ ch^arly later. 'This (h>fines 
d, which we shall (^onseciucMitly designate by d;i. 

It remains therefon^ to (\stiniat(^ tlu^ amount of the total rein- 
f()r(;ement S/l, and then hnd mean \'ahies of / and j. 



32 ■ MEAN REIXFOECEMEXT OF HEAD 

In case of reinforcing rods which are all of them continuous 
over a head Tvdthout laps, the percentage of reinforcement falls only 
slightly below 4 times that at the middle of a side belt; but on the 
other hand were none of them continuous for more than one panel 
and each lap reached bej^ond.the center of the column to the edge 
of the mushroom, the percentage of reinforcement would not be less 
than 7 times that at the middle of a side belt, and to this must be 
added that due to the steel in the radial and ring rods. Thus the 
percentage of reinforcement here may be varied not onh^ by reason 
of the larger or smaller number of laps over each mushroom, but by 
reason of the length of the laps, from perhaps 3.75 to 7 times that 
at the middle of a side belt. For standard mushroom construction 
using long rods, it may be taken conservativeh^ as a 4.25 times that 
at the middle of a side belt. 

It is impossible to make an estimate that T\dll be accurate for 
all cases, but commonly the 8 radial rods of a 20' x 20' panel are 
equivalent in amount to a single 1-1/8'' round rod, or a 1" square 
bar circumscribing the area under consideration, that is to 4 square 
inches of additional reinforcement to be distributed in the width of 
a single side belt. 

The two rings rod, of which the larger is commonlj^ 7/8" round, 
and the smaller 5/8" round, may be taken to increase the reinforce- 
ment of this area by at least one square inch of cross section, giving 
all told some five square inches of cross section additional, equiva- 
lent forty-five 3/8" round rods, or twentj^-one l/2" rods. It thus 
appears that the increased reinforcement from this source reaches 
from 2 to 4 times Ai, and we may safely assume a mean total rein- 
forcement over this area of 

SA = 7.0 A, (40) 

of which the center of action may be pretty accurately stated to be 
at the middle of the third layer of reinforcement rods from the top. 

In the standard design of mushroom floors for warehouses with 
panels about 20' x 20', the mean percentage of reinforcement for a 
single belt ^i being about 0.23%, may be taken b}^ (40) for a rein- 
forcement 7.5 Ai as 

7.5 X 0.23 + = 1.75% The corresponding value of j is 0.83, 

and we shall have 

i 2 A = 0.83 X 7.5 Ai = 6Ai (41) 

As pre\aoush' stated, these equations (containing estimated mean 
numerical values) are given as a specimen computation for the purpose 



STRESSES AT EDGE OF CAP 



33 



of making comparisons. In actual design, computations like these 
should be made which introduce the exact values appearing in the 
design under consideration. 

We now derive from (39) and (40) by the help of (23) the follow- 
ing equations for this area where the belts all cross : — 

5' z ±(1 — K^) q(a + b) 
f^ = Eei = ±Eid3 = (3x^ 



dx' 



90jdsA 



a') 



(1 — K^) 

Ml = 7.5 Ai j dsf, = q (a -{- h) (3x^ — a^) 

12 



..(42) 



in which j and ^3 are less than in (33) and (34), as has been stated 
previously. 

Apply (42) to find the stresses at the edge of the column cap 
on the long side Li. 

Let B = 2x he the shortest distance along the middle of the 
side belt parallel to x between the edges of the caps of two adjacent 
columns, and introduce the values j = 0.83, K = 0.5, and W = 4:qah, 
then; 

TFLi(Li+L2) (35VL?— 1) 1 

f.= ■ 

800 c?3 Ai L2 y 



Ml = 7.5 Aad,f, 



TFLi(Li + L2) (35VLI — 1) 
128 Lo 



(43) 



J 



in which 7.5 Ai is the effective cross section of the steel in this area, 
and Ml is the true resisting moment of the steel derived from the 
elongation, and ^3 is as stated after (39). 

Take the case of a square panel, and assume th(^ diameter of 
the column cap to bo 0.2Li, tluni B = O.SLi and (43) rcHhicc^ to: 



/s 



W Li 



435 d:i Ai 



Ml = 7.ryAiJd.,f, = 



W Li 
70 



(44) 



It will he readily secMi that if ^/3 in (44) is \\\{)vc (hnii 0. I ol \\\v \iM-ti('nl 
distance designatcMl l)y d^ in (34), (as it must \)v) then th(^ str(>ss f^ 
in (31) at i\)v middle of the side belt cxccmmIs /^ in [\'A) at tht> inlgc* 



34 GREATEST STRESSES OVER COLUMN 

of the cap. But this does not prove that the stress in the concrete 
at the edge of the cap is less than that at the middle of the side belt, 
for, the value of i in (37) at the middle of the side belt is about 2/3 
and at the edge of the cap about 1/2, as will be seen by consulting 
Turneaure and.Maurer, page 57, for values of i corresponding to the 
values of j at these points. Hence, using these values of i, if primes 
be used to designate the stress at the edge of the cap, we have by 
(38), /://e = 2/; //, (45) 

from which it is seen that the stress /s at the edge of the cap must 
be only half that in the side belt in order that the corresponding 
stresses in the concrete may be equal. But ordinarily 2/s >/s, and 
so /c >/c. The stress in the concrete at the edge of the cap will be 
computed from that of the steel found in (44) by using (38), in which 
if we put ^ = iT = J, we have /c = /s /15, as the computed value 
of the stress at the edge of the cap. 

Tests have seemed to show that much higher compressive 
stresses may be safely permitted in the concrete around column caps 
where there is compression in two directions, than in the extreme 
fiber of a beam where compression takes place in one direction only. 
A like principle applied to the extreme fiber at the middle of the 
side belt where tension exists at right angles to the compression 
would show that there only a low value should be permitted in 
compression. 

In order to compare the greatest stresses in the steel across the 
mushroom with that at the middle of the side belts in a square panel 
let 5 = Li = L2 in (43), then the stress in a section thru the column 
center along the edges of the panel over the mushroom area is found 
from the following equations: 



/s = 



200 d-i Ai y 

W L^ 



M, = 7.5A,jdsU 



32 



(46) 



which are to be compared with (34), from which it appears that if 
ds in (46) is more than 7 / 8 of c/i in (34) , the stress in the steel across 
the mushroom is less than at the center of the side belts. In any 
case these stresses are so nearly equal that the inadvisability of 
decreasing the steel in the mushroom head below standards indicated 
above is evident. However, some of the steel at the edge of the 
mushroom especially the outer hoop is at such level in this right 



STRESSES OVER CAP 35 

section of the head as possibly to assist the concrete in bearing com- 
pressive stresses. Such a large portion of this section, moreover, 
falls within the cap, that no question of its stability and safety need 
arise, in case the collar band of the column is sufficient to resist the 
comparatively small tensions of the radial rods. 

It will be noticed that in order to make /g and /c as small as 
possible in this area ds must be made as large as possible, i. e., the 
steel at the edge of the cap must be raised as near the top of the slab 
as possible. Neglect of this is to invite failure and weakness such 
as has overtaken certain imitators of the mushroom system. 

A final remark is here in place respecting the values of j and ds 
in this area. The stresses /g and /c diminish very rapidly towards 
the lines of contra-flexure, where they vanish, and the fact that the 
steel also rapidly increases its distance from the top of the slab at 
the same time might be regarded at first thought as requiring some 
modification of the assumptions we have made as to the values of 
j and ds, which are approximately correct at the edge of the cap 
where the steel is placed as near the top surface as due covering will 
permit. But the fact is this: the only consideration of importance 
is the one respecting the position of the steel in that part of this 
area where the moments and stresses are large. The effect of the 
position of the steel near the lines of contra-flexure is negligible, and 
the fact that the amount of reinforcement may be somewhat smaller 
near these lines than elsewhere may also be neglected, so that the 
mean effective reinforcement previously estimated is likel}^ to be an 
underestimate rather than the reverse. Further, the fact that the 
slab is practically clamped horizontally either at the edge of the cap 
or the edge of the superposed column, instead of at its center as 
assumed in our formulas, renders the results given thus far slightly 
too large. 

Good average values of the size of steel used in the standard 
mushroom system of medium span would make the radial rods 
9/S" round, the outer ring rod 7/S" round, the inner ring rod 
5/S" and the belt rods 3/8'' round. The im])ortance of having 
the belt rods small is that for a given thickness of slab i\\c snialUM* 
these rods are the larger is d in both (34) and (43) and couscHiucMitly 
the smaller is/„ and Ai. 

11. In attempting to consider tlu^ stn^ssc^s in (lu^ diagonal rods 
of the central rectangle between the sidi^ l)(»lls of ;i |);iiu^l. it will In* 
noticed, as stated before, that no inw IxMuliiig inoniiMits arc i)r(>|)o- 
gated across the vertical planes or lines of coiitra-fli^xuiH" ['2[) which 



36 DEFLECTIONS IN CENTRAL AREA OF PANEL 

bound it, and since the vertical shearing stresses at these lines are 
uniformly distributed along them, as already shown, (28), there are 
no true twisting moments in these planes. The curvatures of this 
rectangle will consequently^ depend upon its own loading and the 
resistance of its own moment of inertia, regarded as uniformly dis- 
tributed, independently of that of other parts of the slab. 

Hence (21) may be correctly applied to this area, regardless of 
the values which / (and q) may assume elsewhere, provided only 
that the values of I in other areas may be assumed to have constant 
values thruout those areas, and, further, that those areas are sym- 
metrically disposed, so that all central rectangles have one and the 
same given value of I thruout, all side belts also have one given 
value of /, and the mushroom heads have a given value also, each of 
these three sorts of areas being independent. The truth of this 
proposition has been heretofore tacitly assumed in applying (21) 
to these latter areas as has been done. 

It will be seen however, that the values of z obtained from such 
diverse equations express deflections of any point xy on the supposi- 
tion that all the areas considered have the same value of /; but these 
separate equations, each with its own peculiar value of /, can be 
used separately to find the difference of level Zi ■ — ■ Z2 between any 
two points Xi yi and X2 ?/2 which lie in an area where / may be regarded 
as constant. We shall return to this point when we come to the 
derivation of practical deflection formulas. 

For convenience in computing stresses in the rods of the diago- 
nal belt, let the direction of the coordinates be changed so that in 
square panels they will lie along the diagonals which make angles 
of 45 ° with those used thus far. In (21) let 

X = \sl2{x' + y'), y = h^2{x' ~y'), then 

z = ^^ ~ ^^\l ^ [a' - a\x' + y') + x^'^ + l^x^y] .... (47) 
24 E i j d^lA 

in which the panel is square and the axes of x and y lie along its 
diagonals, while the value of ^A /g is the effective cross section per 
unit of width of all the reinforcement in this area regarded as a 
single uniform sheet of metal, and g = 7/8 a, is the width of a 
diagonal belt, and is equal to the diameter of the mushroom head. 
In rectangular panels g = 7/16 (a + 6). 



ELONGATIONS AND SHEARS IN CENTRAL AREA 37 



From (34) we have 
8z (1 -K')qg 



8x 24.Eijd^i:A 



lx'(x^ + 3y'^) — 2a^x'] (48) 



5' z b^z (1 —K^)qg 

ei=e2 = —id—~=—id— = [2a'— 3(x''+2/''](49) 

bx^ by^ 24:EjdI,A 

b^ z (1 — K^) q g x' y' 

and — ^ ; = (50) 

bx by 4.Eijd^i:A 

These expressions satisfy (20) as they should, for (20) is inde- 
pendent of the directions of the rectangular axes x and y. 

From (49) it appears that ei = = /g, on the circumference of 
the circle x^-\-y'^ = fa', which passes thru the points where the 
lines of contra-flexure intersect. 

By (19), which holds for any rectangular axes, and by (50), 
we find 

n' = 1(1 ~ K) q x' y' (26)' 

From (26) it appears that in sections by all vertical planes 
parallel to the diagonals, the twisting increases uniformly with the 
distance from the diagonal. 

Hence by (9) we have 



-S2 = 



( 
( 



5mi 5n'^ 
bx by ) 




5m2 6n ' "^ 

by' bx' } 


1 ^ iqy' 

J 



(28) 



It thus appears that the same law holds for vertical shearing 
stresses on planes parallel to the diagonals, as holds in (28) for planes 
parallel to the edges of the panel. 

In standard mushroom designs the edges of the diagonal belts 
intersect on or very near to the edges of the side belts. That makes 
the middle half of the central square to be ('()\'(m-(hI by (l()ul)li^ b(^lting, 
and the remainder of it by single belthig, so that ^..l = l.o.U or 
perhaps 1.6 A2, and the mean value of A, the nMnforctMuiMit \)cv 
unit of width of shib here, is to be found by dividing this by the 
width of a belt, wliich is 7/8 a. We shouhl then (ind .1 = 1.5 A 2/ 
7/8 a = ].7 A^/d- Hut this mean \ahi(> olM is not its nu^ni etTiu't- 
iv(^ vahie lor this area, because (he reiiiforciMnent is s(> (lisposcul as 



38 MEAN REIXFORCEMEXT STRESS AT PANEL CEXTER 

to furnish the larger values of / in the central diamond just where 
the largest true applied moments and stresses occur. The mean 
value of A in the central diamond is 2^2^7/8 a = 2.3A2/a. The 
mean effective value lies between these two extremes, probably 
nearer the latter than the former. A similar question was discussed 
in connection with (40) and (41). We shall assume as the mean 
effective reinforcement in this central rectangle, 

A = 2A2/a, and / = 2A2 i j dl/a 
or in case of rectangular panels 

I = 4:A2ij dl/ia + h) ....(51) 

In case of rectangular panels the term 2a^ in (49) should be replaced 
by a^ + &^ as a mean value to make it depend the dimensions of the 
panel symmetrical!}", as it must. Making these substitutions in 
(49) we have at .t = = y the center of the panel. 

, W (Li + L2) {L\ + Ll) C, W Li 



1024 Li L2 A2 j d2 256 A2 j d2 

W(L, + L2) {Ll + Ll) C, W Li 



(52) 



Ml = 2A2 j d2 /s = 

512 Ll L2 128 

where Ci = \{Li/L2 + 1) (1 + Li/L?). Take j = 0.89. 

If 1 >L2/Li> 0.75 then 1 <C Ci <^ 1.042, hence Ci varies less than 5% 
while L2/L1 varies by 25% between its extreme permissible values. 
Ci may ordinarily be taken as unity, or may be found with sufficient 
precision by interpolation between the values just given. 

The steps by which these equations (52) were deduced may not 
seem conclusive, since they are not rigorous. They need be only 
good, working approximations for the purpose for which they will be 
here used, viz, to show that the stresses at the center of the panel 
are less than those at the mid span of the side belts in case Ai = A2. 

The value of c?2 in (52) is less than di in (34), but always more 
than 90% of it. We may define d2 as the vertical distance from the 
center of the second and upper of the two diagonal belts to the top sur- 
face of the concrete. We may assume c?2 = 0.9di and j = 0.89 in 
(52), and then we maj^ compare these stresses for a square panel as 
follows : — 

175 

fs = — i; (53) 

205 
where /J refers to the center of the panel. Even were the smaller 



LINE OF ULTIMATE WEAKNESS 39 

value for the mean reinforcement, 1.7 A2/ a, used in deriving (52) 
and (53), the stress given by these equations would not exceed that 
given by (34). The compressive stress /c in the concrete at the 
center of the panel may readily reach a dangerous value in case the 
forms are removed too soon. It should therefore be carefully con- 
sidered in each case. Here, we have an approximate value oi i = 2/3 
and (38) then becomes /c = /s/30 with no possible assistance from 
steel reinforcement since that is all on the bottom of the slab. An 
estimate that the elastic stress in the steel at the center of the panel 
does not much exceed 80% of that at the middle of the side belt 
cannot be far from the truth. 

While this is undoubtedly the fact, it will appear on further 
consideration that local stresses and strains which exist at incipient 
failure are of such magnitude as to make the weakest points of the 
diagonal belts to lie ultimately not at the center, but, instead, just 
outside the diamond where they cross each other. 

Take the standard case where the central diamond reaches just 
across to the side belts. For square panels imagine a circle to be 
drawn concentric with each column of radius L/2. Any circle at a 
column will be tangent to the edges of four diagonal belts across the 
tops of the four columns adjacent to it, and then the octagon cir- 
cumscribing it, whose sides cut at right angles all the belts that cross 
this column head, intersects but a single belt of rods as every point 
of its perimeter. It is evident that, so far as reinforcement is con- 
cerned, such a line or section cuts less steel per unit of perimeter 
than any other regular figure concentric with the column and that 
the reinforcement is entirely symmetrically disposed about the 
column center, so that in case of equal diagonal and side belts, it 
would be impossible from their geometry to distinguish the one from 
the other by anything inside the octagon. That fact would make it 
inherently probably that the stresses and strains of the rods where 
they cross any one side of this octagon should be approximately the 
same ultimately as in those that cross any other side, whether they 
be rods in a diagonal belt or in a side belt. And what will be at- 
tempted to b(^ shown immediately is that ultimately the stresses 
and strains in these several belts approach equality. If that should 
be establish(Hl, it will follow from the conclusion already reached as 
to the excess of the stress(\s and strains of the sidc^ belt over those at 
the center of the panel, that ultimately those at tlu^ edges of the 
ocatgon exceed those in the same rods at the center of the ])anel. 

The (pialification im])lied above in afhrniing that tliis is what 
will ()('(uir ultimately, is for the purpose of conv(\viiig (he ulcii that 



40 LOCAL STRESSES AXD STRAIXS 

this is the approximate distribution of stresses and strains which 
will take place when the slab is sufficiently loaded to bring the steel 
at the middle of the side belt to the yield point. At less stress than 
this there is so much lag in the distribution of the effect of loading 
that it penetrates to the various parts of the slab unequally. 

Taking up now the deferred proof that the diagonal rods where 
they cross the edge of the octagon are subject ultimately to the same 
local stresses and strains as the direct rods of the side belts; note 
that these diagonal rods lie in a triangular area between two side 
belts, which latter experience equal elongations ei in directions at 
right angles to each other. The edges of the triangle in which the 
single layer of diagonal rods lie are continuous with the side belts 
and necessarily experience the same elongations, which are propo- 
gated from the side belts into the triangle by the agency of horizontal 
shears on its edges. Such equal elongations at right angles imply 
the same elongation in every direction in the triangle, as appears 
from the fundamental properties of equal principal stresses and 
strains. Hence we have the same elongations along the diagonal 
rods as along the rods of the side belts at the edges. The existence 
of an ultimate stress and strain in the diagonal belt equal to that in 
the side belt would require that the cross sections A2 and Ai of the 
two belts should be equal, altho so far as the elastic value of /g at 
the center of the panel is concerned A2 might be less than Ai, as has 
been already shown in (52) and (53). The relationships of stress, 
load, etc., for this ultimate condition, have been already given in (37). 

Besides the stresses and strains in the diagonal belts, just in- 
vestigated, those due to the local stretching (arising from the deflec- 
tions themselves) exert their greatest effect on the rods of the diagonal 
and side belts just in the region of the line of weakest section, and 
partly because of that fact. While these local stresses may not exceed 
10% in addition to those already present, their existence should 
prevent any thought of taking 2 A larger than ^i in (37) when deriv- 
ing the ultimate stresses at the 3deld point. Similar results may be 
formulated to cover cases where g is greater or less than 7/l6 L. 

It is perhaps desirable at this point to consider a little more at 
length the matter of local stretching in a slab. It is impossible for a 
continuous flat floor slab to undergo the deflections which we are 
treating, consisting of convexities, concavities, etc., without local 
stretching to allow this to occur. A floor slab of many panels does 
not undergo any change of its total linear dimensions which would 
account for these corrugations. A continuous beam under flexure 
would have its extremities drawn toward each other. But not so 



ACTUAL DEFLECTIONS IN SIDE BELTS 41 

to any such extent with a slab. Such contractions are resisted by 
local circumferential strains which result in true stresses. An 
investigation of such stresses leads to the conclusion just stated that 
in general they cannot exceed 10% of the ordinary stresses due to 
slab bending when they are left out of the consideration. For this 
reason a single panel alone will not function precisely in the same 
way as a panel in a floor of many panels. 

12. Actual deflections are distances which any given points 
of a slab sink down by reason of the application of a given load, and 
their theoretical values are to be computed by help of the formulas 
which have been developed for z in the various areas into which the 
panel has been divided. 

We shall now make a slight modification in our definition of the 
level of the origin of coordinates, and shall take it at the upper or 
lower plane surface of the flat slab before flexure, in which surface 
the axes of x and y are assumed to lie. It is of no consequence 
whether it be the upper or the lower surface which is assumed, the 
equations will be the same in either case. The reason for this new 
definition of the position of the origin is this: Each kind of partial 
area into which the slab has been supposed to be subdivided has its 
neutral surface at a different depth in the slab, and so it does not 
furnish a single suitable level from which to reckon deflections, as 
does the upper or lower surface of the slab. None of the equations 
which have been derived in this paper will undergo any modification 
by reason of this change of definition. It has been assumed that 
each kind of area has a separate value of / which remains constant 
thruout, so that the neutral surfaces of different areas do not join 
at their edges. As previously explained this is of no consequence 
mechanically by reason of the zero true moments that exist at these 
edges. The modification just introduced avoids the geometrical 
perplexities arising from this discontinuity of neutral surfaces. 

Deflections in the side belt area between the lines of contra- 
flexure (24) are to be found from (30), or (31), and (32). To find 
the (k^flection or difference of level in the mid side belt between 
X = 0, y = b, and x = ^ aVs, y = b, substitute these values in (30), 
take i = 0.71, j = 0.91, k = 0.5 and subtract the vahu^ z at the 
second point from that at the first point, which gives th(> following- 
value of the (lefl(>cti()n of the one point below the otlu^*: 

W Li 

A zi = (54) 

10.7 X 10^%/'tyli 



42 ACTUAL DEFLECTIONS IN CENTRAL AREA, IN HEAD 

in which hi is the vertical distance from the center of the single belt 
of rods at the mid span of the side belt to the effective top of the 
slab, considering the strip fill or other concrete finish at its effective 
value. 

In the same manner take the difference of level in the central 
rectangle bounded by the lines of contraflexure between the center 
point Sitx = 0, y =0 and the corner x = ^ a V3, y = ^bVs b}^ using (21) 
and (51) and introducing the values i = 2/3, j = 0.89, etc., and 

C2 = 1ML1/L2 + 1) (1 + LtxLi), then: 

C2 W L\ 
AZ2 = — . ... ,. . (00) 



6.56 X 10^^ dl A 



in which A 2 is the cross section of one diagonal belt and h2 is the 
vertical distance from the center of the upper or second diagonal 
belt to the effective upper surface of the panel at its center. 

On evaluating C2 above, we find 
when 1>L2/Li > 0.75 
then 1> C2 >0.77 

hence we may with sufficient accuracy' for practical purposes assume 
C2 = L2/U (06) 

Deflections in the mushroom area between lines of contraflexure 
(24) are to be derived from (39) (40) and (41) by introducing i = i, 
J = 0.83, k = 0.5 and l^A = 7.5 ^1. Assuming the diameter of 
the cap to be 0.2Li we have, at its edge where x = 0.8a, y = h, from 
(39) 

W Ll (Li/L, +1) /36\2 

z = I — I (57) 

19.1 x lO'Vi Ai Vioo/ 

The value of z at the edge of the mushroom area, where x = f aVs, 
y = h, is to be obtained from (57) by replacing the last factor by 
4/9; and the deflection between the edge of the cap and the edge of 
the mushroom obtained by taking the difference of these quantities 
is as follows: 

W Ll {Li/L, + 1) 

A ^3 = v^. (58) 

60 x 10^^ dlAi 

in which h^ is the vertical distance of the center of the third layer of 
reinforcing rods over the edge of the cap above the bottom surface 
of the slab. 



TOTAL DEFLECTIONS BELOW EDGE OF CAP 43 

Similar expressions may be obtained for the values of z and A2 
on the side parallel to y, where x = a dX y = 0.86, and y = \ 6V^3, 
by exchanging L^ and L2 in (57) and (58). 

Take half the sum of (57) and the corresponding values so ob- 
tained 2X X = a, y = 0.85, as the value of z at the edge of the cap 
where it is intersected by the diagonal of the panel, viz. 

W (Li + U) {L\ + Lt) / 36 \ 2 

z = ( — I (59) 

38.2 X lO^'^ Li L2 dl Ai VlOO/ 



and subtract this from the value of z on the diagonal at the corner 
of the mushroom area where x = \ aV^3, y = 3 6V^3 and we have 

C2 W L\ 

Az^ = --^ (60) 

12.5 X 10'° dl Ai 

as the deflection along the diagonal between the edge of the cap and 
the intersection of the lines of contraflexure, in which C2 and ^3 are 
as previously defined. 

Let Dl = Azi + Azs\ ^^g-^^ 

and D2 = Az2 + Az4^f 
in which Di is the deflection of the mid point of the side belt below 
the edge of the cap, and D2 is the deflection of center of the panel 
below the edge of the cap. 

The proportionate deflections of these points are obtained by 
dividing by the spans, viz: Di / Li and D2 / ^ L\ + Ll. 

13. Estimated proportionate deflections may be obtained from 
(61) under such circumstances as to convey reliable information 
respecting what may be reasonably expected. Let h = the total 
thickness of the slab. The limiting values of the thickness of 
standard mushroom construction are expressed as follows: 

Li/20>/i>Li/35, (62) 

and assuming that the reinforcing rods are l/2" rounds witli 1 2" 
covering of concrete we shall have from the definitions of di, (I2 and 
d:i, already given 

h = r/, + 0.75 = d. + L25 = d.^ + 1.75 (63) 

Substituting these iu (()2) etc. we have 

L,/20 — 0.75 > f/, > /.1/35 — 0.75 ] 

L, /20 — 1.25 > d. > Lx /35 — 1.25 \ (64) 

L,/20- 1.75 > d:, > />, /35 1.75 J 



44 



PROPORTIONATE DEFLECTIONS 



If it be assumed that we are dealing with medium sized panels 
about 20' X 20' (64), may be written in the form: — 

(1 — 0.062) Li/20 > ^1 > (1 — 0.02) Li/35 
(1 — 0.1) Li/20 >d2 > (1 — 0.036) Li/35 
(1 — 0.15) Li/20 > 4 > (1 — 0.05) Li/35 

0.94 di 0.98 

or, > — > 

20 Li 35 

0.90 d2 0.964 

> — > (65) 

20 Li 35 

0.85 ^3 0.95 

> — > 

20 Li 35 



In (54), (55), (58) and (60) replace W Li by its value given in (34), 
viz, 175 di Aifs, and we have 



A ^1 = 



A Z2 = 



A ^3 



A ^4 = 



Llfs 



6.11 X 10^ di 

C2d,L\A,f, 



3.75 X 10^ dl A2 

d^Ll{L,/L2 + l)/s 

34.3 X 10^ dl 

C2 dl Li /s 

7.14 X 10^ dl 



(66) 



in which /g is the greatest stress in the steel, ^. e., at the mid side 
belt, employed here to express deflections instead of expressing them 
in terms of panel load as was done previously. 



RELATR^E DEFLECTIONS AT MID SPAN 



45 



Introduce into (66) the numerical values given in (65) which 
will then express limiting values of deflection for medium spans. 
For simplicity let Lj = L2 then: 

287 > > 170 

162 > > 106 

10' A^2 I (67) 

660 > > 451 

10'A^3 

275 > > 188 

1O^A04 



By (61) we have the proportionate deflection of the side and diagonal 
belts as follows: — 



L287 66OJ 

[- + -1 

Ll62 275J 



< < — + 



10' 

/a 



Li 



f - ^ -1 

Ll70 451 J 



/s 



/s 



—. — r < ; < 

10' V2 Li V2 

/s 



L1O6 188j 



/s 



10^ V2 



200 X 10' 



< < 

D2 
< < 



123.4 X 10^ [ 

/s 



141.4 X 10' Li V2 95.9 X 10^ 
If f, = 16000, 



1 D2 1 
< < — 

884 Li V2 600 



If f, = 24000, 



If /„ = 32000, 



1 D2 
< 

590 Li V2 



< 



400 
1 



1 Do 

< T < 

442 /., V2 300 



(68) 



(09) 



Larger spans then 20 ', or smaller Mov\ tluui 1 2" romul, or Aj^/.i 
will reduce the above values soinewliat , wliile smaller s})aus or 



46 THEORETICAL AND EXPERniEXTAL DEFLECTIONS COMPARED 

larger steel will increase these values, all of which can in each case 
be submitted to calculation b}' the methods here developed. 

To recur at this point to the expression for the deflection D2 in 
terms of the panel load W by help of {00), (60) and (61) 

C2 TT' ^' 

10^° 

By (65) we find 



Ai L6.06 dl 12.5 dl\ 



90 do 96.4 
— > — > 

85 d^ 95 

and using this inequalit}^ to eliminate c/3 from (70) we find after 
reduction 

C9 W L\ Co W L\ 

< Do < 



4.46 X 10'° r/i Ai ' 4.33 x 10'° dl A^ 

from which we may write as a mean value 

Co W L\ 

D2 = " ,. , ^ (71) 

4.4 X 10^° dl Ai 

The empirical deflection formula given on page 29 of Turner's Con- 
A^rete Steel Construction, when written in these units, is 

W Ll 

D2 = 7^—r- (72) 

4.84 X 10'° di Ai 

This is identical with (71) when Co = 0.909, and diverges from it 
slightly for other admissible values of C-o. The practical agreement 
of (71) and (72) affords a second confirmation of the theoretical 
deductions made thus far, and this taken in conj miction T^4th the 
practical identity of formulas (34) and (35), the theoretical and 
empirical expressions for the maximum tensile stresses in the rein- 
forcement, furnishes what on the theory of probabilities ma}^ be 
regarded as so strong a probability of the general trustworthiness 
of the entire theory as to exclude any rational suppositition to the 
contrary. 

The A'arious formulas for stresses and for deflections which have 
been developed in this paper have been obtained under the express 
pro^dso that the panel under consideration was assumed to be one 
of a practically unlimited number of equal panels constituting a 
continuous slab, all of which are loaded uniformly and equally. The 



MUSHROOM PANELS INDEPENDENT 47 

question at once arises as to the amount and kind of deviations from 
these formulas which will occur by reason either of discontinuity of 
slab or loading, such as occurs at the outside panels of a slab or at 
panels surrounded partly or entirely by others not loaded. The 
answer to this question depends very largely upon the construction 
of the fiat slab itself. 

In the standard mushroom construction it has been found that 
the stresses and deflections of any panel are almost entirely inde- 
pendent of those in surrounding panels. This is due to the fact 
that the mushroom head is an integral part of the supporting column 
in such a manner that it is impossible for it to tilt appreciably over 
the column under the action of any eccentric or unequal loading of 
panels near it. When single panels have been loaded with test 
loads, no appreciable deflections have been discoverable in sur- 
rounding panels, and no greater stresses and deflections have been 
discovered than were to be expected in case surrounding panels were 
loaded also. Future careful investigation of this maj^ reveal 
measureable effects of this kind, but they must be small. 

A like statement cannot be made of other systems of flat slab 
construction where the reinforcement over the top of the column is 
not an integral part of the column reinforcement itself. Tests on 
these systems have shown clearly the effects of the tipping of the 
part of the slab on the top of the column, and lack of stiffness of 
head, in the increase of the deflection of the single loaded panel over 
the deflection to be expected in case of multiple loaded panels, and 
especially in the disturbance of the equality of the stress in the other- 
wise equal stresses in the rods of the side belts. Such distrubance, 
by increasing the stress in part of these rods, would necessitate larger 
reinforcement in the side belts of such s^'stems than would be re- 
quired in mushroom slabs. The great stiffness of the nmshroom 
head is also of prime importance in taking care of accidental and 
unusual strains liable to occur in the removal of forms from under 
insufficiently cured slabs. 

14. In considering the (h^sign of the ring rods and radial 
cantilever rods of the mushroom head, it should be borne in mind 
that they occupy a position in sucli close proximity to thi^ \v\'v\ 
of the neutral surface as to prevent tlu^ni from bcMUg s\il)i(M't(Hl 
to severe tensile or compressive stresses by r(>ason of \\\c l>eniling 
of the slal) as a whole. Theii' |)rin(*ipal function as slab nuMU- 
bers is to resist shearin<>: stress(»s and (he b(M\(hnir s(r(>sst\s (\uv to 



48 VERTICAL SHEAR .\ROUXD COLUMN CAPS 

local bending. Their total longitudinal stresses are too small in 
comparison to require consideration. 

Let a cylindrical surface be imagined to be di'a^TL concentric 
with a colunni to uitersect the slab, then the total vertical shearing 
stress which is distributed on the surface of intersection is ecjual to 
the total panel load TT' diminished by the amount of that part of the 
panel load hdng inside the cylinder. If the cylinder be not large, 
the total shear may be taken as approximately equal to TT itself. 

It is e^ddent that the smaller the diameter may be that is 
assumed for this cylinder, the greater ^dll be the intensity of the 
vertical shear on its surface and that for two reasons: First, because 
the tot?l load thus carried to the colunm A^dll be greater the smaller 
the diameterj and second because the surface over which the total 
shear vnll be distributed decreases vriih. its cUameter. 

The result of this is that the dangerous section for shear is the 
cylindrical surface at the edge of the cap. For cylinders smaller 
than this the increased vertical thiclaiess of the cap chminishes the 
intensity of the shear. We proceed therefore to consider the manner 
in which the total vertical shearing stress of approximately W in 
amount is distributed in the material of the cylindrical surface at the 
edge of the cap. 

In a beam or slab the horizontal shearing stresses due to bending 
reach a maximmn at the neutral surface. It is a fundamental con- 
dition of eciuilibrium that shearing stresses on planes at right angles 
shall be equal, and it is this condition that determines the distribu- 
tion of the vertical shears, which are at right angles to the horizontal 
shears resulting from bending the slab as a whole. From this we 
have the well known fact that the A^rtical shear varies from zero 
at the upper and lower surfaces to a maximum at the neutral surface, 
and this is necessarily the manner in which the total shear is dis- 
tributed at the edge of the cap. The top belt of rods "^dll be sub- 
jected to comparatively small shearing stresses, and successive 
layers of rods ^dll be tmder larger and larger shearing stresses bj^ 
reason of their greater nearness to the neutral surface, while the 
total shear borne by the radial rods near the neutral surface '^dll be 
much larger than that upon the others. The shearing stress in 
the concrete will need to be considered also. 

It is to be noticed that all the steel of the belts and mushroom 
head act together without the necessity of supposing large com- 
pressive stresses in the concrete to transmit vertical forces, because 
the belts of reinforcement rest directly upon each other, and these 

in tinii upon the ring rods and radial rods, all in metallic contact 



VERTICAL SHEAR IN RADIAL RODS, ETC. 49 

with each other, in the mushroom head, and so they transmit and 
adjust the distribution of stresses within the system to a very large 
extent independently of the concrete. 

We can then safely assign moderate values of the shearing 
stress to each of the elements that constitute the slab at the edge 
of the cap, with the assurance that they will each play a part in 
general accordance with the distribution which has been already 
explained. 

The mushroom is constructed of great strength and stiffness 
not merely to effect the results which have appeared previously in 
the course of the investigation but also to ensure the stability of the 
slab in case of unexpected or accidental stresses due to the too early 
removal of the forms, before the slab is well cured, at a time when 
the only load to which it is subjected is due to the weight of the 
structure itself. 

The working load to be assumed in designing the mushroom 
may be taken as the dead load of a single slab plus the design load, 
provided sufficiently low values of the shearing stresses be assumed 
in the cross sections of steel and concrete at the edge of the cap 
for the support of this working load, as follows : 

For slabs having a thickness of /i = L/35 a mean working 
shearing stress of 2000 lbs. per square inch at the right cross section 
of each reinforcing rod which crosses the edge of the cap, a mean 
shearing stress of 40 lbs. per square inch in the vertical cylindrical 
section of the concrete at the edge of the cap, and 8000 lbs. per 
square inch of right cross section of each radial rod. 

For slabs having a thickness of /i = L/20 the intensities just 
given may be safely increased by 50 per cent for reasons that will 
be explained later. For slabs of intermediate thickness increase 
the intensities proportionately. 

These values are sufficiently low to enable the structure to sup- 
port itself before the concrete is very thoroughly cured, and the 
head so designed will be found after it is well cured to be so j^ro- 
portioned as to carry safely a test load of doubU^ tlie livc^ and dc^ad 
loads for which it was designed. 

In this connection it seems desirable to invest igat(^ what takes 
place in case of overloading and inci]Ment failure of an insufhciently 
cured slab, or one unduly weakened by thawing of ])artially froztMi 
concn^te. Suppose that under such circumstancc^s a, shearing crack 
were formed extending completely thru (he liend at iUv vdix.^' of tl\e 
cap, and we wish to inv(\stigat(^ tlie stn^ssc^s and h(^h:i\ior o( {\\c rods 



50 STRESSES IN RADIAL RODS 

that cross the crack at which shearing deformation has begun to take 
place. Designate the position of the crack by X. 

The total vertical shearing stress on a radial rod at X is the 
sum of two parts found as follows: First, the vertical reaction at 
the top of a column is made up of the vertical reaction of the con- 
crete core of the column and the reactions of its vertical reinforcing 
rods. Call the vertical reaction of one of these rods Fi. The rod 
is bent over radially and Vi expresses also the amount of the vertical 
shear in that rod where it starts out radially from the column. 
Between this point and X for a distance which measures usually 
from 9 to 12 inches, the rod experiences the supporting pressure of 
the concrete in the cap under it to a total amount which we will 
designate by V2. The total shear in the radial rod at X will then 
amount to 

V = Vi + V2 (73) 

provided we neglect the weight of that small part of the actual load 
of the slab which lies directly over this piece of the rod and may 
be regarded as resting upon it. This portion of the radial rod of 
length I is a cantilever fixed at one end in the top of the column, and 
carrying a load V at the other end with a supporting pressure under- 
neath of total amount V2 whose intensity is greatest at X and gradu- 
ually decreases along I from X to the fixed end. The rod has a 
point of contraflexure and zero moment at X. The portion of the 
rod outside the crack has a fixed point in the slab at the place where 
it supports the inner ring rod, at a distance from X which should 
not exceed I as just defined. Similar conditions hold for this leng-th; 
i. e. there will be a totol shear in the radial rod at a point just inside 
the inner ring, rod due to its total shear outside this ring rod and to 
the vertical loading imparted to it b}^ the ring rod itself. To this 
must be added the doA\Tiward pressure of the concrete between the 
inner ring rod and X. All these, together, constitute the total 
shear — V at X, in equilibrium with the reaction + V already ob- 
tained at that point. 

We shall discuss separately the action of V\ and V^ upon a radial 
rod. A load Vi at the end of a cantilever of lenglh I causes a 

deflection of amount ^1 = J Fi f / EI (74) 

in which 1= tt ^^/64 where ^ = the thickness of the rod. 

Also Vi = siA , A= TT^V^ 

in which Si = the mean shearing stress per square vmit of cross sec- 
tion and A is the cross section of the rod. Hence 

si = ?,ZiEt^ /mf (75) 



STRESSES IN RADIAL RODS 51 

which shows that so far as Vi is concerned, for an}^ given displace- 
ment Zi the shearing stress carried per square unit of rod will be pro- 
portional to the square of its diameter, and up to its permissible 
limiting shearing resista,nce, each unit of section of such a rod will 
be effective in proportion to the square of its diameter. For econ- 
omical construction, this will require the radial rods to be few and 
large, rather than numerous and small. The bending moment is 
greatest at the distance I from X and amounts to Vi I. The stress 
in the extreme fiber due to the bending moment Vi I in the rod is 
Pi = Vi lt/2I = 8si l/t (76) 

This equation shows that the stress in the extreme fiber is so very 
large at the fixed end of the rod compared with the shear at X that 
so far as Fi is concerned the rod will suffer permanent deformation 
by bending long before there is any danger of its shearing. Vi is so 
large compared with V2 that this conclusion will not be altered when 
we come to consider the combined action of V2. 

Incipient failure of this kind will therefore cause distortion and 
sag without collapse. In case such sag as occurs in this case is 
detected underneath the head around the cap, the slab should be 
blocked up at once and the concrete picked out at all parts showing 
facture. This should then be refilled with a stronger concrete 
which will set rapidly. Such repair should not weaken the slab. 

Whenever the intensity with which a radial rod presses upon 
the concrete at the edge of a crack at X passes the compressive 
strength f,, of the concrete, it must begin to yield. At this instant 
we shall have a pressure of the concrete against the rod which gradu- 
ally diminishes as we pass along the rod from X to the distance /, 
where it becomes zero. We shall assume that the pressure dimin- 
ishes uniformly with this distance. This may not be precisely cor- 
rect, but cannot be much in error. If the shear V2 at X is the sole 
cause of this pressure, then ^2=2 ^ ^/o and I V2I = ^ t t~ f^. 
is the bending moment in the rod at the distance /, due to V2 at X 
and the pressure distributed along /. 

It will be found that these produce a deflection 

^2 = '^fj'/20FA = 0.3/'^ V2/EI (77) 

a unit shear of 

.S2 = V2/A = Z2E t^/4.S f (78) 

and a stress on the extreme (iber at. a distauc(> / aiuoiiiilini;- to 

p,^ = y.^ 1 1 /si = i().s,. i/t (79) 



52 SHEAR IX OUTER RLN'G RODS 

It thus appears that the equations expressing the aotion of V2 
are precisely similar to those for T^i, differing only in their numerical 
coefficients, and consequently all the statements as to the resistance 
of the radial rods under the action of Vi hold for the action of 
T^i and T'2 together in the case of given initial deformations, 
Zi = z-i at X. 

While the prececUng investigation has. in order to make ideas 
explicit, ostensibly assumed a crack at X and an initial small shear- 
ing deformation at X, the investigation applies equally well to the 
elastic shearing deformation of the concrete at the dangerous 
section in which case the total shearing stress ^ill consist of an addi- 
tional componenent due to the resistance of the concrete, which 
however may for additional safety be neglected. If the assumed 
deformation be confined T^dthin limits so small that the concrete 
is able to endure it without cracking then the preceding investiga- 
tion may properly be applied to it. It is right here that the thick- 
ness of the radial rods is able to render its most effective ser\dce, 
for it appears from (75) and (78) that any permissible intensity of 
shear may be developed in the radial rods by making them of suit- 
able thickness, even tho the deflection be kept within the elastic 
limits of the concrete. 

As already stated we must not overlook the fact that the major 
stresses here are those under the head of T^j. which are due to the 
direct metallic contacts of the steel rods "resting one upon 
another, where large stresses are transmitted and pass independ- 
ently of the concrete except for the distortions of the steel which 
meet resistance, and the secondary reactions such as have been 
treated in a single aspect while iuA^estigating the action of T^2- 

It is due to this fact that large shearing stresses may be safely 
borne by the slab at and near the edge of the cap. which the concrete 
mostly escapes, it merely furnishing some lateral stiffening to the 
steel. On this principle the outer ring rod should have a cross 
section not much less than one half that of the radial rods on which 
it rests. For, this arrangement provides for the transferal to the 
radial rods of all the shear the ring rod is able to carry, it being in 
double shear compared '^^^th the radial rod it rests on. 

It is impossible to determine the cross section of the inner ring 
rod, with the same definiteness as that of the radial rods, but 
that is unimportant. Its position has already been fixed as not 
more than I from the edge of the cap, where I is the distance from 
the top hoop or collar band of the column to the edge of the cap. 



STRESSES IN CONCRSTE OF HEAD 53 

The vertical shearing stresses may be regarded as sufficiently 
resisted outside the mushroom by the concrete alone. The critical 
cylindrical surface separating those areas where the shear may be 
assumed to be safely carried by concrete alone, from those areas 
where the steel may be relied on to carry as much of the shear as 
may be required, should evidently be taken somewhat inside the 
outer ring rod, but just where is of no particular consequence. 

The supposition of the existence of a crack at X, either actual 
or potential, on which our computation of the stresses in the radial 
rods has been based, is sufficiently satisfactory so far as the rods 
themselves are concerned; but it seems desirable to consider in more 
detail the phenomena attending the development of the stresses in 
the concrete at and near the edge of the cap, especially in soft con- 
crete when the limit of its compressive resistance is reached in this 
region. 

The horizontal compressive resistance of the concrete at the 
lower surface of the slab is that already treated in (38) , and it is our 
present object to consider how that is to be combined Avith the 
vertical supporting pressures under the radial rods, and with the 
horizontal and vertical shears in the slab due to bending. These 
latter are greatest in the neutral surface, as has been previously 
stated, and according the general theory of stresses are equivalent 
to, and may be replaced by, a compression and a tension in the mate- 
rial respectively at 45° with the vertical (and mutually at right 
angles) of the same intensity as the shear. It is evident that the 
combination and resultant of these three compressive stresses 
would form the dangerous element in the stress, since the single 
tensile element would be relatively unimportant, and it would find 
assistance in its resistance from the steel running in a direction thru 
the concrete such as to afford it substantial support. This direction 
is that of the straight lines on the surface of a right cone whose 
vertex is above the center of the column and whose slope is 1 to 1 . 

Consider now two of the elements of the compression in the 
concrete around the cap, viz, the horizontal compression which is a 
maximum at the lower surface and zero at the neutral surl'a('(\ and 
that due to sliear which is parallel to tlu^ sidt^s of a right vouv witli 
vertex downward, whose sides have an u})ward and outward slope 
of 1 to 1 , while its int(^nsity is so distributed that it is y.vvo at the 
bottom of th(^ slab and greatest at the neutral surfa('(\ It ap})t\irs 
consequently that the liiu^s of gn^atest c()ni])r(^ssion in the concn^tt^ 
due to the combination of these two (>l(Mn(Mits of conipnvssiou wouKl 



54 COMPRESSION ON CONCRETE OF HEAD 

lie in vertical planes on a bowl or saucer-shaped surface that is hori- 
zontal at the edge of the cap and inclined at a slope of 45"^ at the 
neutral surface ; and if the concrete were to crush under these stresses 
alone, the surface of fracture would have the shape indicated in- 
stead of that of the cylindrical surface previously assumed. This 
change would not, however, materially affect the computations we 
have made of stresses in steel; it merely serves to fix more definitely 
the position of the points of contra-flexure of the radial rods. 

But there is still one further element or component of the total 
compression in the concrete to be considered and combined with 
those just treated in order to arrive at the resultant or total com- 
pression. This componenent is that due to the concentrated press- 
ures underneath each of the radial rods. These rods are at some 
distance apart circumferentially and so do not exert a pressure that 
is uniformly distributed circumferentially. Any concentrated stress, 
such as that in the concrete supporting a rod, diffuses itself in the 
material in such a manner that its intensity rapidly diminishes with 
the distance from the surface of the rod, in accordance the same law 
as exists in case of centers of attraction. Since the supporting com- 
pression under the rods is vertical, we can imagine the lines of great- 
est compression in the concrete, when this component is combined 
with those already mentioned, to lie in vertical planes on a bowl or 
saucer-shaped surface which has as many indentations or scollops 
around its edge as there are radial rods, at which indentations the 
slope of the sides is such more nearly vertical than a slope of 45°. 
At such parts of the surface the intensity is also more severe, and 
especially is this the case if the slab is thin so that the concentrated 
pressure has small opportunity to distribute itself by radiating into 
a considerable body of material before it reaches the bottom of the 
slab. It thus comes about that thick slabs are enabled to carry 
safely larger intensities of shearing stress around the cap than can 
thin slabs, which is in accordance with and in justification of the 
statements already made as to permissible shears around the cap. 

The resulting surface of fracture due to shear and compression 
around the cap would be of irregular conical shape starting from 
the edge of the cap and extending thru the entire thickness of the 
slab, were this not interfered with in the upper part of the slab by 
the mat of reinforcing rods, which are so tenacious as to tear to 
pieces and fracture the upper surface to a considerable distance in 
all directions whenever any such fracture occurs around the column. 

Nevertheless such fracture as here described does not under 



RADIAL AND RING RODS PREVENT FRACTURE OF HEAD 55 

any ordinary circumstances result in a dangerous collapse of the 
slab, or one that cannot be repaired without much difficulty, for, the 
radial rods and the reinforcing rods will at most have suffered some 
individual deformation by bending and are still far from being 
broken. This will become evident later where an experimental 
attempt to load a full-sized slab to failure is described in detail, and 
full account of the results reached is explained and illustrated. 

It is stated on good authority that in experience with many 
hundreds of buildings constructed on this system, no case of shear 
failure or even of incipient shear failure or fracture has occurred in a 
well cured slab near the column and while a few cases of incipient 
failure have occurred in floors where forms were prematurely re- 
moved, no injury or fatality has resulted therefrom to any person. 

It appears chat the line of weakest section in the cured slab of 
the standard mushroom type is that discussed previously in obtain- 
ing (37) and shown in Fig. 3 page 7. This is brought out later by a test 
to destruction of a fairly well cured slab. The line of weakest sec- 
tion in a partly cured slab is on the other hand not definitely fixed, 
but may be and sometimes is, shearing weakness near the column as 
has been discussed and pointed out. Provision against such weak- 
ness or carelessness is a safeguard which, while costing a small 
amount in the matter of steel, is an insurance against serious acci- 
dent well worth the investment involved. It is secured by making 
the radial and ring rods sufficiently stiff and strong. 

15. This section will be devoted to a consideration of the 
mushroom system, and to several more or less similar flat slab 
systems, in order to comment on the modifications in mechanical 
action that are produced by the particular modifications of the 
arrangement of the reinforcement in these systems. 

Fig. 1, page 2 represents the section of a standard mushroom 
head by a vertical plane thru the axis of the column. In this the 
elbow rods are shown, the vertical portions of which arc embedded for 
such distances as may be necessary in the columns or are them- 
selves column rods. One of these is represented separately at the 
right side of the Fig. They are confined just undcu* the elbow at 
the top of tlui column by a steel neck band, and are bent ovi^r at 
the elbow to extend radially into the slab. This Ix^ut o\cv ])ortiou 
is formed to scale as to length and sl()j)es in accordance witii the 
size and thickness of the slab in whicli it is to be used, in such a 
way that when the ring rods and four layiM's of slab rods rest 
upon it and an^ tied in pla('(% (he (op of the iip])(M- laA'cr will be 



56 STANDARD MLSHROOM SYSTEM 

0.75 inch below the top of the slab at a distance of the thickness 
of the slab outside the edge of the cap, and at the same time the 
extremities of the radial rods '^'ill be 0.5 inch above the bottom of 
the slab. In order to accomplish this, the radial portions of these 
rods must be nearly horizontal over the cap, and have a suitable 
slope outside the cap as shown in Fig. 1. 

Fig. 3, page 7, shows the ground plan of the reinforcement of the 
mushroom slab when the panel is square so that Li = L2 = 2a 
= 26. In this Fig. the diameter of the mushroom head is assumed 
to be of the extreme size g = L/2, sl size which would increase the 
cantilever beyond that in usual practice to an extent not adopted 
except in the case of A'ery miusual intensity of loading. It ^*ill 
be observed that the areas where tlie reinforcement consists of a 
single belt or layer are thereby rendered small, and the slab action 
due to the mutual lateral action of belts which cross each other 
exists over nearly the whole slab. 

In Fig. 2, the dimensions of the rectangular sides are so taken 
that Li Li = 0.75, which is assumed to be the limiting or smallest 
value of that ratio for constructional purposes. Further, the 
diameter of the mushroom is made as small as will permit the rein- 
forcing belts to cover the entire panel, viz. ^ = 7 (a + 6)/l6. For 
example if Li = 20, and Lj = 15, we haAT g = 7.65 + . This may 
be considered to represent standard practice, where the edges of 
the diagonal belts intersect on the edges of the side belts. This 
was the case assumed for treatment in deriving the formulas of the 
preceeding investigations. Those formulas could be modified to 
apply to larger values of g. by taking lines of contra-flexure at the 
edges of the head nearer the panel center than given by (24). and 
by taking larger values of tlie effective cross section of steel than 
those employed in (32), (40) and (51). 

Xow it is evident that systems similar to this may differ from 
it in several ways : — 

1st. The design of the frame-work at the top of the coluron 
ma}' be different from this ^dthout any change in the belts of re- 
inforcing rods. It is hardly possible for any other form of frame- 
work to be substituted for this which will exhibit the same rigidity 
of connection between it and the colunni as do the elbow rods 
embedded in the column and bent over radially in the slab so as 
to make the colunm and slab integral with each other by means 
of this common reinforcement. Any reduction of the stiffness of 
connection between column and frame-work of head results in in- 
creased tipping of the head under eccentric loading of the slab. 



OTHER SYSTEMS 



57 




Fig. 4. 

Eccentric loading is any loading of one panel differently from 
another. Tipping of the head increases some deflections at the 
expense of others, and increased stresses in some of the reinforcing 
rods at the expense of others, and so requires some additional 
reinforcement. Such a frame-work is illustrated in Fig. 4, which 
merely rests upon the top of the column without the support of 
metallic connection with the vertical column rods. It consequently^ 
affords less resistance to tipping under eccentric loads than when 
stiffened by such metallic connection. 

2nd. The ground plan of the reinforcing belts may remain un- 
changed but part only of the belt rods may be carried at the top 
of the slab over the column head, while the rest of them are carried 
thru under the head at the bottom of the slab. This modification 
of design, when a sufficient number of rods go over the head to 
resist the negative bending moments there, is very uneconomical 
of steel, because in the case where they all go over the head, it is 
the fact that altho the mean tension of the steel is not so great as 
at mid span, nevertlu^less, by reason of the overla])ping of the belts 
in crossing, the stresses in the* rods at the to]) reach a value not 
much less than at mid si)an, and cannot be saft^ly diminishtHl in 
number. It thus appears that the rods carried thru on tht^ bottom 
are largely sujx^rfluous. Of these two mats of rods at top and 
bottom, on(; of them is necessarily in tension and tlu^ other in com- 
prefjsion. But it is a mistake to us(^ Mvc\ to n^sist conipn^ssion 
when concrete can ])e l)etter used for tliis |)iirpos(\ The lo\vi>r mat 
is superfhious for this reason. 



58 



SMALL HEAD. TOP AND BOTTOM BELTS 







— Theoretical line \ 

Column tread A^ of inflection ^ 
/ ' ^Assumed line 



of inflection 



I 1 1 M I 



W in pounds per square foot 



I I 



I ^,58^*' • - • "uji/ago/ia/ steel in tuo layers ' ' ""'^^^ 






\ 1 y^ rectangular sttel In 
^^' ' ''o H y^ two layers 






^^ / 


5 


In pounds per linear 


^<"" L_, Fig. 5 



There is still another and, if possible, more serious objection 
to this arrangement of rods to form a mat or double laj^er of rods 
at the top and at the bottom of the slab near the columns. This 
is because they are too far removed from each other in the slab 
for the elongations of the steel in one mat to be resisted hj lateral 
contractions in the other. The reinforcement does^not therefore 
conspire to produce the slab action expressed by Poisson's ratio, 
which requires that the interacting steel concerned should lie approxi- 
matel}^ in the same zone or level. 

This arrangement is illustrated in Fig. 5, copied from Taylor 
and Thompson's Concrete Plain and Reinforced, p. 484. In this 
design the size of the head is small enough to reduce the width of 
the belts so greatly that not onh^ are the areas where we have a 
single layer of rods on the plan much enlarged, but we find that 
nowhere do more than two layers lie in metallic contact with, each 
other, and the areas where even this occurs are limited to one 
relatively small square over each column, and one of equal size 
at the middle of each panel. The remaining areas are subject to 
the law of single rod reinforcement, where we must assume lateral 
action to be such as greatly to diminish K for the combination, a 
fact very injurious to the efficiency of the reinforcement. This 
as has been said, is due partly to the smallness of the head and 
partlj^ to the separation of the laj^ers between the top and the 
bottom of the slab. 



BAD EFFECT OF ANY SHARP BEND OR ELBOW IN A ROD 



59 



3rd. Another modification of design without change of ground 
plan is that where the rods that are carried over the head at the 
top of the slab are given a sudden steep dip at the line of contra- 
flexure to carry them to the bottom of the slab at that line. This 
is also illustrated in Fig. 5. Such sudden bends or kinks any- 
where in the rods may give rise to very serious fractures because 
of straightening out under tension, especially when the forms are 
removed. Such bends give rise to great differences of stress in the 
extreme fibers of the rods, thus diminishing their resistance also. 
All sudden bends in rods embedded in concrete should be sedulously 
avoided as tending very effectively to crack the concrete, whether 
the rods are part of the belts or in the frame-work of the head, as 
shown in Fig. 3, in which are many such angles and elbows unsup- 
ported except by concrete, and therefore objectionable. 

It seems fair to conclude that the cracks shown in the plan 
of the floor of the Deere & Webber Company Building, Minnea- 
polis, tested by Mr. Arthur R. Lord, and occuring along the edges 
of some of the loaded panels at the upper surface, where none usually 
appear, were due to the elbows in the frame work of the head, like 
that in Fig. 4, in conjunction with the comparatively small resis- 
tance to bending in a vertical plane offered by the rods forming this 
projecting elbow. 

In the mushroom head the only bend permitted is that at the 
elbow of the radial rods where a strong steel neck band prevents 
any such bad effect as has just been pointed out. 




FiR. 6 



60 TWO WAY REINFORCEMENT 

4th. We may notice a form of design in which the diagonal 
belts are omitted and the entire panel is covered by rods parallel 
to the sides of the panel. This, while apparently very different in 
ground plan from those just considered does not differ from it 
materially in principle. It is clear that the lattice pattern of the 
web in this case is in many parts of the panel not woven so close 
as where diagonals exist, while in other parts of the mesh the num- 
ber of layers in contact with each other has been decreased. Experi- 
mental results do not as yet enable us to determine with certainty 
whether Poisson's ratio for this combination is as great as for the 
mushroom. Upon that depends in part the relative efficiency 
of the two arrangements. A form of this design is seen in Fig. 6. 

The maximum deflections at the center of a loaded panel of 
the system of Fig. 6, would occur when the panels touching its 
four sides were also loaded. In this particular it differs from a 
loaded panel in a mushroom slab which would theoretically have 
its deflection slightly decreased by loading surrounding panels, 
tho this is too insignificant to have been observed as yet. 

Deflections shown by tests of this system of two way reinforce- 
ment are wholly inconsistent with simple beam theory, and can 
only be explained on the basis of slab theory. Nevertheless, some 
of its advocates attempt to design its reinforcement and com- 
pute its strength on the basis of beam theory, which actual de- 
flections show to be untenable. Such attempts should be entirely 
abandoned as erroneous and misleading. 

All considerations which have been discussed under the three 
previous counts are to be taken as applying equally to this plan 
of arranging the reinforcing rods, especially as to carrying of 
part of the belts thru on the bottom surface at columns. 

5th. Another element of design is the relative number of 
rods in the side and diagonal belts. We have previously adduced 
reasons to show that in a square panel the same number of rods is 
required ultimately in the diagonal belts as in the side belts, tho 
for stresses less than the yield point of the steel, it would be pos- 
sible to diminish the number of rods in the diagonal belts some- 
what. Equation (34) shows that for equal stresses in the steel 
of the side belts the number of rods should have the same ratio 
as the lengths of the sides. 

A different rule from this has been erroneously proposed, 
viz., that the ratio of the number of rods in the side belts should 
be equal to the ratio of the cubes of their lengths. The only foun- 
dation for this rule is that according to the beam strip theory as 



RELATIVE CROSS SECTION OF BELTS 61 

developed in Marsh's Reinforced Concrete, p. 283, a rectangular 
plate carried by a level rigid support around its perimeter, would 
divide the load per unit of area which is carried by two unit-wide 
rectangular strips that cross each other, as the fourth power of their 
lengths, and hence would carry to the edges of the rectangle loads 
proportional to the cubes of the lengths of those edges. Were this 
so, the case of a horizontal rigid support around the entire peri- 
meter of the panel is wholly different from support on columns 
at the corners, and such a rule would be wholly inapplicable there- 
fore to a floor slab so supported. This rule was, however, evidently 
adopted in the design of the Larkin Building, Chicago, as shown 
by a photograph of its reinforcement in place before the concrete 
was poured, to which the writer has access and published in Cement 
Era for February, 1913. The very exhaustive tests of this build- 
ing made by the Concrete Steel Products Company of Chicago, 
and published in the Cement Era, for January 1913, show that this 
ratio of rods caused the stresses for the larger loads to be more 
than twice as great at the middle of the short side belts as at the 
middle of the long side belts. This was assuredly an uneconomical 
distribution of steel, since correct design would require these stresses 
to be equal, when in fact one exceeded the other by 120 to 140 per 
cent. This discrepancy would be largely rectified by making the 
number of rods directly proportional to the lengths of the sides, 
as required by (34). 

It also appears that the diameter of the mushroom head and 
the width of belts of slab rods in the Larkin Building is less than the 
limiting size in the standard mushroom system, viz. g = 7{a-\-b)/lQ. 
This makes the intersection of the diagonal belts fall nearer the 
center of the panel than the edges of the side belts. The very 
considerable effect of a very inconsiderable change of this width 
has been mentioned on p. 25. The result would be that the steel 
would for this reason be far less effective, and its resistance 
would be more nearly in accordance with (37) than with (34), a loss 
of perhaps 25 to 30% in its effectiveness. 



62 SPECIMEN COMPUTATION OF THIN SLAB 

16. This section will be devoted to a specimen computation 
applying several of the preceeding formulas to a floor slab of practi- 
cally the same dimensions and reinforcement as one or two recently 
designed and now under construction (1913). 

Long Side Li = 28' X 12 = 336'^ 
Short side L2 = 25' 10'' = 310". 
Thickness of rough slab, h = 10" = L/33.6. 
By (56) C2 = L2/L1 = 0.9 nearly. 
Diameter of head g = 1 {L^ -\- L2)/32 = 141". 
Diameter of cap L^—B = 0.2Li = 67". B = O.8L1 - 268.8". 
Each belt has 25 — 7/l6" round rods. 

Cross section of each belt^ A = 25 x 0.15+ = 3.76 sq. inches. 
Depth of center of mid side belt with J inch concrete cover- 
ing, di = 10 — 0.5 — 0.2 = 9.3". 

Depth of center of second layer slab rods at panel center, 
d2 = 10 — 0.5 — 0.64 = 8.86" 

Depth of bottom surface below third layer of slab rods at edge 
of cap with M" covering, 6^3 = 10 — 0.75 — 1.1 = 8.15". 
Design load per square foot = 150 lbs. 
Dead load per square foot = 130 lbs. 
Panel load, TF = 280 x 28 x 25 5/6 = 202,550 lbs. 

A maximum tension is found in the slab rods at the middle of 
the long side belt, and is to be computed from (34) as follows: 

202550 X 336 

/g= = 11,120 lbs. per sq. inch (80) 

175 X 9.3 X 3.76 

Any other loading within elastic limits of the steel would 
produce proportionate stresses. 

The tension in the steel at the center of the panel is com- 
puted by (52), as follows: 

1.02 X 202550 x 336 

fa= = 9,145 lbs. per sq. in.. (81) 

256 X 3.76 X 0.89 x 8.86 

The radial tension at the edge of the cap is by (43), 

202550 X 336 x 646 (3 x 0.64 — 1) 

/s = = 5320 lbs. per sq.in . (82) 

800 X 8.15 X 3.76 x 310 



COMPUTED STRESSES IN THIN SLAB 63 

The circumferential tension at the vertical section thru the 
center of the coluron at the end of the long side may be computed 
by placing B = Li in (43), and we obtain, 

202550 X 336 x 646 x 2 

/3= = 11,570 lbs. per sq. in. . (83) 

800x8.15x3.76x310 

as the mean computed intensity of stress in each of these rods, 
regardless of its distance from the center of the column. This 
stress may be reduced by increasing the number of laps over the 
head. The result in (83) is, however, an over-estimate of the 
tension across the top of the head because the head is integral 
with the cap of the column where compressions in the concrete are 
no longer confined merely to the thickness of the slab but take 
in a much greater depth of concrete in the cap. This in effect 
puts the neutral surface at a lower level throughout the cap and 
by thus increasing the lever arm of the reinforcement reduces its 
tension and deformation. This will react upon the rest of the 
reinforcement in such a manner as practically to make the stresses 
smaller than given by (83) because the mean lever arm will have 
increased. In fact the greatest stress in these rods will be that 
given by (80), instead of (83). 

The compression in the concrete leng-thwise of the longer 
side belt at its middle is to be computed from (38) and (80) as 
follows: By taking the percentage of belt reinforcement at 0.3%, 
the corresponding value oi i = 0.72, and E^/E^ = 15: 

0.28x11120 

/e= = 288 lbs. per sq. in (84) 

0.72 X 15 

The compression at the center of the panel where the per- 
centage of slab reinforcement may be conservatively assumed at 
0.6% and i = 0.66 may be computed thus: 

9145 

/c= = 305 lbs. per sq. in (85) 

2 X 75 

The compression at the edge of the cap lengthwise of the 
side belt is uncertain in the absence of exact information as to the 
laps in the slab rods over the head. Assume that one-half the rods 
arc lapped over each head, and that we take six belts as the reinforce- 
ment of the slab, the percentage then is 1.8% and i = J, then, 

5,320 

/c= = 355 lbs. ])or s(t. in (SO) 

15 



64 COMPUTED DEFLECTION OF THIN SLAB 

For reasons already given in discussing the circumferential 
tensions in the head, it appears that any computation of the cir- 
cumferential compressions in the concrete on the basis of (38) would 
be incorrect and subject to large errors of as much possibly as 
50%. That this is the fact appears evident when we consider the 
large mass of concrete in the cap which must be actually diminished 
in lateral dimensions before the slab which is integral with it can 
be subjected to true stresses of equal intensity, and consider also 
that near the edges of the head the radial rods and the outer ring 
rods approach the lower surface sufficiently to afford reinforcement 
to resist compression. It is consequently unnecessary to look 
further than (86) in computing the greatest compression in the 
concrete. 

As previously stated, computations based on (38) are highly 
artificial and arbitrary in their character, since they assume the 
straight line theory as well as an arbitrary value of the ratio of 
Young's moduli for steel and concrete. Furthermore, concrete 
in compression in both circumferential and radial directions at the 
same time, as it is at the edge of the cap, is known to resist with 
safety compressive stresses of greater intensity than when in simple 
compression in one direction. 

If a test load of twice the design load, viz., in this case of 300 
lbs. per square foot, be placed upon the slab, the deflections which 
will be produced by the addition of this total load of 217,000 lbs. 
may be computed as follows: 

217000 X 336^ 

By (54), A ^1 = = 0.237 .... (87) 

10.7 X 10^° X 9.3^ X 3.76 

0.89 X 217000 X 336^ 

By (55), AZ2= = 0.378 .... (88) 

6.56 X 10'° X 8.86^x3.76 

217000 X 336^ x 2.084 

By (58), A ^3= 7, -. = 0.115.... (89) 

60 X 10'° X 8.15^ x 3.76 

0.89 X 217000 X 336^ 

By (60), A ^4 = -^ 2 = 0-235 • • • ■ (90) 

12.5 xl0'°x8.15^x 3.76 



PROPORTIONATE DEFLECTIONS 65 

By (61), jDi - 0.352, and D2 = 0.613 (91) 

Di 1 D2 I 

= , and - 7 = (92) 

Li 960 V^^2 _^ ^^2 745 

Any loading differing from this would produce deflections 
proportionate to its intensity. 

In this specimen floor slab, which is near the limit of least 
thickness permissible in the standard mushroom system, viz., 
d = Li/35, it is clear that the design load brings stresses to bear 
upon its reinforcement which are very moderate in their intensity 
indeed. It is also evident that were the slab to be loaded with a 
test load of such amount that the total load sustained would be 
twice the dead load of the slab itself plus twice the design or live 
load, viz. 560 lbs. per square foot, none of the steel would be stressed 
up to the yield point, and the first failure would take place by 
cracking the concrete, tho the steel would still prevent sudden 
failure and collapse. Altho the slab is relatively so thin the de- 
flections are also very small for so large a span. 

It has not yet been so generally recognized as it should be 
that a thin construction, such as a flat slab is, should not be ex- 
pected to show so small proportionate deflections as is required 
in girders. 

The observed results of quite a number of tests of mushroom 
slab floors are to be found on pp. 32 and 44 of Turner's Concrete 
Steel Construction. These are there compared with results com- 
puted according to Turner's empirical formula, which translated 
into our present notation has been reproduced in equation (72). 
The observed and computed results show a very close agreement. 
The results given by (72) are in close agreement, as has been seen, 
with those derived from (61). 

Some of these test slabs present peculiarities of reinforce- 
ment such as need to be individually considered in order to make 
exact computations of their deflections. It is thought that the 
specimen computation aln^ady given will afford sufficicMitly guidance 
in the methods to be emi)loye(l. 

Having considercnl the stresses and (U^flections of a slab whicli 
is near tlu; minimum thickness for i\w standard mushroom system, 
viz. L,/35, it will \)v instnuiive to consicUM' a s])ecini(Mi or two 
near the maxinuim thicluu^ss />,/20. 



66 



STIFFNESS OF MUSHROOM SLAB BRIDGES 




Tischers Creek Bridge, Duluth 




Test of Tischers Creek Bridge with 30 ton construction cars, each loaded with 20 tons of rails 
Deflection less than one twenty thousandth part of the span 



SPECIMEN COMPUTATION OF THICK SLAB 67 

Take for example the bridge over Tischer's Creek, Duluth, 
shown in the cuts on page vii and page 66. It is supported on three 
rows of columns crossing the gorge, at a distance apart of 27 feet 
center to center of columns, the two street car tracks being over 
the side belt that lies along the center line of the bridge lengthwise. 
Each of these rows consist of six columns lengthwise of the bridge, 
at a distance apart of 26 feet from center to center, so that 

Li = 27 X 12 = 324" 
L2 = 26 X 12 - 312" 
The size of the mushroom heads and width of the belts is 12 feet, 
which is in excess of 7 (Li + L2)/32 = 139 I/8" = 11.6', thus giv- 
ing great stiffness. The object to be obtained by maximum thick- 
ness and large head is to secure great stiffness and so reduce vib- 
rations as well as decrease deflections. There are twenty 9/l6 
inch round slab rods in each belt, or a total cross section in each 
belt of Ai = 5 square inches of metal. The slab is 15" deep at 
its thinnest part at the gutter on each side of the roadway, and 
the steel is kept down to that level throughout the slab, altho at 
the crown of the roadway under the tracks and over the center 
row of columns the slab is 5" thicker, or 20", with the same thick- 
ness over the side rows of columns where the sidewalks are. The 
mean thickness is somewhat in excess of L2/2O. This makes 
di = 19" for the short side belts, di = 17" for the long side belts 
and ^3 = 14" approximately for the heads. The design load per 
square foot = 150 pounds. The dead load of the slab per square 
foot = 300 pounds. Hence IF = 450 x 26 x 27 = 315,900 pounds. 
The effective cross section of slab steel is so great by reason of large 
heads that instead of (34) we may take 

W L 

f.= (34) 

200 di Ai 

For the long side belt this gives /« = 6,033 pounds per square inch. 
The total load imposed on the slab might be made six times as great 
without causing the steel to reach its yield point, and the live 
load might become 900 pounds per square foot w^ithout causing /^ 
to exceed 16,000 pounds. 

This slab was tested as shown in the cut, page 66, by ninuini:; 
two construction cars loaded with 20 tons of rails eacli over the 
})ridge at the same time along one track of tlu^ short side b(^lt 2() 
feet long. Weight of each car = 60,000 ])()un(ls. \\'eiglit of rails 
40,000 pounds. Total weight of train = 2()(),()()() ])ovin(ls extend- 
ing over several spans. The deflections was loo small to Ix^ dis- 
covered !)y ()])servatioiis with level miuI rod. it is us(>h^ss to nttcMupt 



68 



COMPUTATION OF THICK SLAB 



to compute the deflection of this slab under the test load because 
the four steel rails of the railway tracks across the bridge were so 
fastened to the steel cross ties which were embedded in the con- 
crete as to make the rails a part of the reinforcement of the slab. 
They furnish a cross section of reinforcement equal perhaps to 
7 Ai, which would effectually bar the application of our deflection 
formulas and reduce deflections to very small quantities. 

In so thick a slab as this the action of any contemplated load 
is widely distributed by the slab itself, and such loads, as well as 
all shocks and vibrations are largely dissipated or absorbed by the 
body of slab itself without causing observable local stresses as they 
do in steel structures. 




VIEW OF REINFORCING STEEL 



Flat Slab Bridge, Denver, Colo. 



Spans 43 ft. 6 in. 



Carries Heavj- Interurban Cars 



COMPUTED STRESSES AND DEFLECTIONS 69 

The Curtis Street bridge, Denver, Colorado, is one of four 
bridges across Cherry Creek, shown by the cut on page 68, con- 
structed on the mushroom system It has three rows of three 
columns each crossing the stream, the middle column of each row 
in mid stream with spans of 42 feet between columns centers length- 
wise of the bridge, thus obstructing the waterway as little as pos- 
sible. It has a width of 28 feet between column centers. The 
slab is 17 inches thick at the gutters, 26.5 inches at the sidewalks 
outside the gutters, and 21'' over the center row of columns. The 
sidewalk is stiffened with fourteen 3/8'' round rods lengthwise 
just below its top surface as supplementary reinforcement, and 
there is an outside parapet giving added stiffness. There are 
also three stiffening rods 24" apart across the bridge midway 
between columns. There are three ring rods, and the width of the 
belts is 16'. This is in excess of 7 (Lj + L2)/32 = 183.75" = 15 5/l6'. 
The heads are exceptionally stiff each having twelve 1 3-8" round 
radial rods. Each belt has twenty-six 5/8" round rods, hence 
Ai = 26 X 0.3 = 8 square inches nearly. 

Li = 42 X 12 = 504" , L2 = 28 x 12 = 336". 
The dead load per square foot = 300 pounds. 
The design load per square foot = 150 pounds. 
IF = 450 X 42 X 28 = 529,200 pounds. 
di = 20" for long side belt. 

Compute the stress in the steel by (34) modified to (34)' by 
reason of exceptional stiffness, and we obtain f^ = 13,320 pounds. 

Compute the central deflection due to a test load of 100 pounds 
per square foot. Let ds = 16". Then in (71) L2/Li = 2/S: hence 
€2= 3/4, and we have D2 = 0.125". This is probably considerably 
in excess of the correct deflection, since the slab is stiffer than the 
one considered in equation (71), which was derived for 20 foot spans. 
More correct values are to be computed from (54), (58) and (61). 
Moreover for such comparatively light stresses in the concrete, 
the deflections, as we hav(i seen previously fall short of those com- 
puted by the formula, whi(^h agrees with ex})eriment for stresses 
nearer the yield point of the steel. Do = 0.125" is less than 
one four-thousandth of the span, and th(^ deflection under {\\v 
working load would undoubtedly be less than o:w sixlh-lhoiisandth 
of th(^ span. 



70 WORKING STRESSES AND FACTOR OF SAFETY 

A word is here in place respecting working stresses and the 
factor of safety in the reinforcement of slabs, to the effect that 
the same values of these quantities in slabs affords a greater degree 
of security than in ordinary structural steel construction, and that 
occurs for several reasons: 

1st. Steel rods such as are used in slabs have a higher yield point 
by perhaps 25% than the steel of other structural members. Fur- 
thermore, it is quite possible and desirable to use a higher carbon 
steel for these rods than the mild steel necessarily used in structural 
work, where it must be manipulated in such ways that high carbon 
steel cannot be used. But in these rods which suffer no usage 
tending to impair their condition, there is good reason to use a steel 
of higher yield point and greater ultimate strength. This yield 
point may readily be 70% greater than that of ordinary mild steel 
for structural purposes. 

2nd. Rods embedded in concrete do not yield as do bare 
single rods in a testing machine or elsewhere by the formation of 
a neck and drawing out at that point. The concrete embedment 
prevents that. 

3rd. In a reinforcement consisting of multiple parallel rods 
acting together, no single rod can become overstrained and yield to 
any appreciable extent before bringing into play adjacent rods. 
This makes the construction tough, and not liable to sudden col- 
lapse, as well as obviates concentration of stresses thus ensuring 
a high degree of security. 



COMPARATIVE TEST OF TW'O SLABS 71 

17. This section will be devoted to a detailed consideration 
of a test to destruction of two slabs, 12' x 12' between column 
centers, constructed for experimental purposes. The tests were 
made by Professor Wm. H. Kavanaugh, in November and December, 
1912, and the results he obtained, together with a mathematical 
discussion based upon them, will be here given. One slab was 
constructed in accordance with the plans and specifications of the 
U. S. Patent No. 698,542 issued to 0. W. Norcross for a slab for 
flooring of buildings, and the other was a Turner Mushroom slab 
under U. S. Patent No. 1,003,384. The test serves to bring out in 
a striking manner not only how two slabs, which present a super- 
ficial resemblance in the plan of arrangement of reinforcement, 
differ from an experimental and practical standpoint, but it also 
makes evident their radical divergence of action mechanically and 
mathematically. 

That two slabs of the same span, thickness and amount of 
reinforcement should on test show that one of them was more than 
twenty times as stiff, and more than five times as strong as the 
other, and that the failure of the weaker one was a sudden and 
complete collapse, with little or no warning to the inexperienced 
eye, while the other gave way by slowly pulling apart little by 
little, thus gradually getting out of shape without any final break 
down, are phenomena that deserve the close attention of the de- 
signer, and are of the highest interest scientifically as well as practi- 
cally. The enormous differences in the deflections and in the 
stresses in the reinforcement as shown by extensomoter measure- 
ments, and in the character of the failure in respect of safety- and 
its relation to the line or zone of weakest section, as well as in the 
difference of design loads and breaking loads amounting to 500%, 
all illustrate what scientific design will accomplish and what results 
are possible by an ingenious arrangement of the reinforcement. 

These slabs were each of the same thickness, viz 6'', and were sup- 
ported by columns placed at the corners of a square 12' x 12' from 
center to center of columns. The slabs projected 2' to 3' beyond 
the centers of the columns on each side, and had precisely the same 
number and size of reinforcing rods in each belt, viz elevcMi 3/8 
inch round rods. The concrete was of a 1 : 2 : 4 mix, and while 
only about four weeks old at the time of the test, it had becni poured 
warm and kei)t warm by steam heat under such unusually favorable 
conditions as to have l)ecome well cured at the time of the test. 
The steel us(mI sIiowcmI by test a stress at yield i)oiut of 51,000 to 
55,000 i)()un(ls ])er scjuan^ inch, and an ultimate strength o^ 70,000 



72 BEAM THEORY, VERSUS SLAB THEORY 

to 80,000 pounds, Tsdth an elongation of twenty to twenty-five per 
cent. 

The first slab was made in accordance vdth. the specifications 
of the Xorcross patent already referred to except that belts of rods 
were substituted for the netting mentioned by the patentee. This 
design was selected as one of the two for this comparative test, 
not because it is a good design, or one that any engineer would 
to-day care to employ, but because it exhibits, according to the 
express intention of the patentee, simple tension on its lower surface, 
everyrv^here between columns, and simple compression everywhere 
on its upper surface between columns; this being in direct contrast 
to the other design, which is arranged not only to resist direct ten- 
sions over the supports, which the first does not, but also to resist 
circumferential stresses both around the supports and around the 
panel centers, as any truty continuous flat slab must. 

This test may then be viewed in the light of an experimental 
demonstration of the difference between a reinforced flat slab con- 
structed in accordance vrith the beam theory and one constructed 
in accordance with correct slab theory, where true and apparent 
moments differ radically as shown at the beginning of this investi- 
gation, but are wholly contradictory to any form of simple or con- 
tinous beam theory. This test may be regarded as settling once for 
all the question of applying simple beam theory to a cantilever flat 
slab, reinforced throughout practically its entire area with a lattice of 
rods crossing each other and in contact. It shows that it is impos- 
sible to compute the deflections of such a slab by beam theory. 
Furthermore this impossibility makes it certain that the stresses 
in such a slab cannot be computed by beam theory, for to do this is 
to commit an inconsistenc}^ such as has heretofore too often been 
committed, but one which should hereafter be carefully avoided. 



THE NORCROSS TEST SLAB 



73 



Norcross in his patent already referred to describes his con- 
struction as consisting ' 'essentially, of a panel of concrete having 
metallic network encased therein, so as to radiate from the posts 

on which the floor rests The posts are first erected, and a 

temporary staging built up level with the tops of posts. Strips of 
wire netting are then laid loosely in place on top of the staging .... 
The concrete is then spread upon or moulded in place on the staging 
to enclose the metallic network. In practice I have sometimes 
laid the concrete in layers of different quality, the lower layer of 
the floor which encloses the wire being laid with the best concrete 

available If the forces acting upon a section of flooring 

supported between two posts be analyzed it will be found that the 
tendency of the floor section to sag between its supports will cause 
the lower layers of the flooring to be under tension while the upper 
layers of the flooring will be under compression, these stresses being, 
of course, the greatest at the top and bottom layers, respectively." 




Fi^. 7. Itciiiforcciiu'iit of N'tircross Slab 



74 



THE NORCROSS TEST 




Fig. 8. Norcross Slab Carrj-ing Load 3 
^'- ^' J. 6"'' O' 0. 6^'-/^' 




Co/. Cop P/o/e 20^20-d 
3di^ l/~8 '^ each u/qy 



Fig. 9. Norcross Slab 



LOADS ON MUSHROOM SLAB 



75 



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76 



NORCROSS SLAB 



The number and arrangement of the reinforcing rods in the 
Norcross experimental slab, (eleven 3/8'' round rods in each side 
and diagonal belt) is clearly shown in the view of Oct. 31st, Fig. 7, which 
shows the forms ready for pouring the concrete. Steel plates 
20" X 20" X 0.5" carry the rods and rest on the tops of the columns, 
which last in this case consisted of steel pipes about 5^" in dia- 
meter filled with concrete and embedded at their lower ends in large 
concrete blocks. A vertical central bolt in the concrete at the 
upper end of each pipe permitted the plates to be firmly secured to 
the tops of the columns. The view of Nov. 30th, Fig. 8, clearly 
shows the manner of placing the pig iron on the slab for load 3. 
This slab is 16' x 16'. The loading at first covered an area having 
the form of a Greek cross whose central square was five feet on a 
side with arms 5' 6" long, as represented in accompanying diagram 
of loaded areas A, B, C, D, E, Fig. 9, and of amounts shown in 
Table 1. 




Fig. 10. Collapse of Norcross Slab 

When 10,000 pounds had been piled on the central part of the 
slab in addition to load No. 4, of 66,812 pounds, the slab suddenly 
failed. In anticipation of such failure timber blocking had been 
placed under the slab to prevent its falling more than possibly ten 
or twelve inches. 



NORCROSS TEST 



77 




Fig. 11. Collapse of Norcross Slab 

The two views of Dec. 2d, Fig. 10 and Fig. 11, show the con- 
dition of the slab after removing part of the final loading in order 
to render the nature of the failure visible. Careful extensometer 
measurements of the elongations of the steel rods at the middle 
of the side and diagonal belts were made under the action of loads 
1, 2, 3 and 4, and also similar extensometer measurements in the 
concrete both on the top and the bottom of the slab along the center 
line of the side and diagonal belts near those edges of two of the 
steel plates which were nearest the center of the belts. Besides 
these, certain other measurements of the concrete were made at 
right angles to the diagonals. Deflections were also measured 
under these loads at the middle of the diagonal belt and of two of 
the side belts at V, W, X, Y, Z. 

These measurements all show beyond question that the side 
and diagonal belts act like simple beams in this form of construction, 
since the stresses in the steel and concrete on the under side of 
the slab in the direction of the rods is invariably tensile, wliilt^ the 
stresses in the same directions on toj) of the slab luv always com- 
pressive. It was the avowed intention of Norcross to reinforce 
the slab in this manner sin(H^ he regardcMl tlie u])])er part of the slab 
as being subjected everywhere to ('()mi)ression and the lower ])art 
to tension only, as stated in liis specifications as aln^ady (quoted. 



78 COMPUTATION OF THE TEST 

The following computation, Table 2, shows a good approxi- 
mate agreement of the results of this test with the beam theory of 
flexure, assuming for simplicity that the stiff steel supporting plate 
and interlacing of the ends of the belts diminishes the effective 
span of the side belts by 12'', and the diagonals in the same pro- 
portion, and further assuming that the loadmg was all applied at 
the middle of the side and diagonal belts. 

The extensometer measurements made were for a length of 
8'', consequently the stress in the steel per square inch would be 
computed thus: 

f^ = l/S (elongation in 8") x 30,000,000; (l)i 

and, this being known from observation, it ^^'ill be possible to com- 
pute the load TT^ carried b}' the beam in which the given elongation 
occurs, as follows: 

The bending moment due to a concentrated load TT' at the mid- 
dle of a beam of length L is M = i W L, (2)i 

and the equal moment of resistance of the reinforcement b}' which 

it is held in equilibrium is M = A j d f^ (3)i 

in which A is the total cross section of the steel in the belt = 
11 x 0.11 = 1.215 sq. in., and the distance from the center of 
the steel to the center of compressive resistance of the concrete 
is assumed to be, j d = 0.9 x 5.75 

when d = 5.75 is taken as the distance from the center of action 
of the steel to the top of the slab, 

Hence W = ^ A j d fjL (4)i 

is the load required to cause the stress /=; in the steel. In the side 
belts we assume the span L to be 132''. and in the diagonals 132 V2. 

In Table 2, which follows, it vr\\\ be noticed that loading Xo. 1 
is too small to develop sufficient elongations or deflections to 
overcome the initial compressions in the concrete in which the 
reinforcement is embedded, so that the load carried by the steel is 
only about one half of the actual load, the other half being evidently 
carried by the concrete in which it is embedded. This is in com- 
plete accord T\ith other similar experiments. But in case of loads 
Xo. 2 and Xo. 3, where the steel is stressed close to the j'ield 
point, the sum of the loads as shown by the stresses in the steel 
is very close to the total actual load. It is assumed that these 
total actual loads are carried by the various belts in the same pro- 
portion as the computed loads, since there is no other waj' of 
dividing the total load between the belts. This may be stated 
mathematically, as follows: 



LOADS AND DEFLECTIONS OF NORCROSS SLAB 



79 



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00 

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t^ t^ l> 

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03 Fl 



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03 



80 NORCROSS SLAB ON THE BEAM THEORY 

Let TFi = the computed load on a side belt, 
and W2 = the computed load on a diagonal belt. 
Let TFi = the actual load on a side belt, 
and W2 = the actual load on a diagonal belt. 
Then 4TFi + 2TF2 = total computed load on slab, 
and 4TFi + 2W2 = total actual load on slab. 

4 TF; + 2 W2 Wi W2 
Then = = — (5)2 

4.W1 + 2W2 Tf 1 W2 

from which Wi and W2 can be computed, Wi , W2 and 4TFi + 4TF2 
being already known. 

The stresses in the steel under load No. 4, are so far beyond 
the yield point as to make computation useless. Having found 
the actual distribution of loading W[ and W2 the center deflections 
of the belts have been computed by simple beam theory from the 
formula. 

W' L^ 

D2 = (6)1 

48E Ai j (f 

in which i d = the distance from the steel to the neutral axis and 
the value of j has been assumed to be 0.69; W is the actual load on 
the belt and L is its span as previously stated. 

It appears from Table 2, that the effect of the reinforcement 
is accounted for to a reasonably close approximation by consider- 
ing the belts to act as a combination of simple beams, at least with- 
in the range of loading near the yield point of the steel. 

It appears that the steel reached its yield point under a total 
load on the slab of from 15 to 18 tons and final collapse occured under 
a total load of a little over twice the latter amount not distributed 
uniformly but piled more in the general form of a pyramid. 

It was observed that the application of the relatively small 
loading on the corner areas F, G, H, I, had a very injurious effect 
upon the slab, tending to break it across the tops of the columns. 

The results of the test may be summarized in the Norcross 
system as follows: 

1st. This slab is of the simple beam type, and the test shows 
no cantilever action and no circumferential slab action. 

2nd. The narrow belts running diagonally leave large areas 
without reinforcement, and there is consequently no provision for 
resisting circumferential tensions as required in slab action. 

3rd. The concrete showed compressive stresses on the upper 
surface of the slab in the direction of all the reinforcing rods. 



SUMMARY OF NORCROSS TEST 81 

4th. The concrete showed tension at the bottom surface in 
the direction of all the reinforcing rods, in agreement with Norcross' 
own analysis. 

5th. This slab deflected 1.6'' under 33 tons and then broke 
down completely under 38 tons. 

6th. The first crack appeared under a load of 15 tons and 
deflection of 0.7''. 

7th. The slab, not being reinforced on the top surface over 
the columns, inevitably cracks at a column when the slab is loaded 
around the column. 

8th. At failure the steel had passed its yield point. The 
percentage of reinforcement in the diagonal belt if we regard the 
belt as about 18" wide is very nearly 1%, but since a width of 
concrete somewhat greater than that may be assumed to act with 
this steel, the percentage of reinforcement is somewhat less than 
1%. Similarily, the side belts of width 36" have a reinforcement 
less than 0.5%. The full strength of the steel in both belts was 
developed by the concrete, which fact demonstrates that the con- 
crete was of high grade and well cured. The steel was also of 
good standard quality, and the test was therefore in every way 
fair to the Norcross slab, since it was so loaded as to cause the 
stresses in the side and diagonal belts to be practically equal, thus 
using the steel most economically. The slab failed because the 
steel yielded near the middle of the spans, thus causing the concrete 
above the steel to crack and break. 

The second slab was made according to the Turner Mush- 
room System, under the patent already referred to. 

Since all forces in a plane may be resolved into components 
along any pair of axes at right angles to each other it is possible 
to provide reinforcement to resist any horizontal tensile stresses 
in the slab by various arrangements of intersecting belts of rods at 
zones where these stresses occur. The combination of such belts with 
radial and ring rods to constitute a large and substantial canti- 
lever mushroom head at the top of each column affords a very 
effectives and economical arrangement for controlling the distribution 
of the stresses in the slab, and it places the reinforcement where 
it is most ncHHled. It not only has the same kind of advantage 
that the continuous cantilever beam has ovvr i\\v siinpU^ girdtM- 
for long spans, l)ut combines with it the kind of su]KM'iority tliat the 
dome has over the sim])le arch by reason of circumferential stresses 
call(Ml into play, whicli greatly adds to tlu^ carryiup; capacity of th(^ 
slab. 



82 



REIXFORCEMEXT OF MUSHROOM TEST SLAB 




Fig. 12. ReinforcemeDt of ^.lushroom Slab 



£ '^ ^ 




/V 



Column Rods 6-/^^ 

Fig. 13. ^Mushroom Slab 



Rin4j Rods ^ ^ 



THE MUSHROOM SLAB TEST 



83 



The mushroom test slab was six inches thick, and was sup- 
ported on four 18'' by 18" square reinforced concrete columns 
distance 12' from center to center. These had square capitals, 
42" X 42". The slab was appromimately 18' x 18', and the dia- 
meter of the outer ring rod of the Mushroom was 66", while the 
inner ring was 42". These were supported on eight 1-1/8" round 
radial column rods. 




Fig. 14. Mushroom Slab, Loud 4. 

This will be clearly understood from the view dated October 
31st, Fig. 12, which shows the reinforcement and forms ready for 
pouring the concrete. The remaining views are explained by their 
accompanying legends. 

The diagram of loaded areas for the mushroom slab Fig. 13. is 
like that already given for the Norcross slab in every particular 
except that the size of the mushroom slab being 18' x 18', while the 
Norcross slab was 16' x 16', the arms of the Gnn^k cross in the 
nmshroom slab are each 5' 6" lonsr and 5' wide. 



84 



LOADS ON NORCROSS SLAB 



o 

ft 



5q 



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o 

02 



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O 



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03 



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per 
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1— 1 1— 1 00 '— 1 T— 1 cc 
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LOADS ON MUSHROOM SLAB 



85 



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86 



LOADS ON MUSHROOM SLAB 




Fig. 15. Mushroom Slab, Load 7. 




Fig. 16. Mushroom Slab, Load 9. 



COMPUTATION OF MUSHROOM TEST 87 

The accompanying Table 3, exhibits the loads per square foot 
of each of the subsidiary areas shown in the diagram as also the 
total loads on each of those areas. The view of Dec. 3, Fig. 14, 
shows load 4, and that of Dec. 13, Fig. 15, load 7, while that of 
Dec. 16, Fig. 16, shows load 9. 

Elongations of steel were measured by Berry extensometers 
in two of the side belts and in one of the diagonal belts until the 
yield point of the steel was reached at load No. 8. Deflections 
were also measured. In Table 4, these will be considered so far as 
they relate to the middle points of the belts. Loads 8, 9, 10, are 
of great interest as exhibiting the behavior of the slab under ex- 
cessive loads, showing, as they do, yielding and large permanent 
deformation without dangerous collapse. 

By (52) the uniformly distributed load per square foot of 
panel area when the stress in the diagonal belt is /g is found for a 
square panel from the expression 

256 j d2 A 

144g = w = TF/144 = f, (52a) 

144 L 

which applied to this slab gives us 

256 X 0.89 X 5.125 X 1.215 

w = /, = /s/14.6 (52b) 

144 X 144 

The values of this uniformity distributed load w is tabulated 
in table 4, for each of the observed values of the /s in the diagonal 
belts. The values of w so computed tend to become identical, 
in case of the heavier loads, with the loads per square foot on the 
central area C, as might reasonably be expected, w; being the uniformly 
distributed load which is equivalent so far as the stress on tlie dia- 
gonal belt is concerned to the action of the actual loads which are 
not uniformly distributed. 

How compute by (54), (55), (58), (60) and (61), the deflections 
at the mid side belt and at center of the panel, due to a uniform load. 
These results are given in Table 4, and accord closely with those 
actually observed, as they should, because the irr(\gulari( y of dis- 
tribution does not produce deflections that diflVM' nuich iVoin the 
equivakuit uniform load as c()m])ut(Ml above. 

In these C()m])utati()ns it is assuiiKul thai di = 5.5", (/o = 
5.125'', d. = 4'' 



DEFLECTIONS OF MUSHROOM SLAB 



The double set of values under loads 4 and 5 is due to the 
fact that readings were had under load 4, immediately after the 
load was appHed, and again 7 days later before applying load 5. 
The second set of readings were the larger as shown. The second 
set of readings under load 5, were taken four days subsequently 
to the first set. 

It appears from Table 4, that the observed results are account- 
ed for by the slab theory to a good degree of approximation 
up to the yield point of the steel. 




Fig. 17. Comparative Deflections of Norcross and Mushroom Slabs. 



A graphical representation of the experimental observations 
in the deflections at the points V, W, X, Y, Z, of the two slabs is 
found in Fig. 17, which shows in a striking manner how small the 
loads and how great the deflections were in the Norcross slab on the 
one hand, and how large the loads and how small the deflections 
were in the mushroom slab on the other hand. 



LOADS AND DEFLECTIONS OF MUSHROOM SLAB 



89 



a 

o 

>5 

S-i 

o 

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13 -. CO 

a g.2 

a; ^ 

CO J^ 03 
Q^ ^ (D 

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Pi -Q 

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CD ^ 

CO O) 

T5 CO 
o3 ^ 
O O 

<V 





GO 


O CO CO l> 




00 


c 


00 CC 






O 


lO 1— 1 !>• T— 1 


LO 


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t^ 


t^ 


lO GO lO LO 


r- 


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o 


ZO y-t ■ • 


o 


oo 




o 


(M 


o 


CM 




00 


O O O Oi 


1-H 


O T-H CO 




(N 


LO T— 1 LO CO 


GO 


O CO LO 


CO 


CO 


LO CO • '^^t^ 


CO 


LO CM CM 


o 


CO 1—1 


o 


lO 




o 


(M 


o 


CM 




CO o 


o 


T— 1 t^ 


O CM 




t^ i-H 


o o 


o c 


O CO 


lO 


CO CO 


OOOOOOOO^GO 


'vH CO 


O t^ O 00 CO ^ 


o o 


TtiOiLOtOOGOOO 


o e 


LO CM ^ CM t^ rtH 




o o 


T-HCMOiLOCO^CO'* 

CM ^ . . . . 


o c 


1—1 1—1 1—1 CM T-H CM 






CO CN 










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OOCOCMt— ILOt— It— 1 


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mid 


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.B 
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ition 
(61) 
obs. 




6 

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Obs. elong. of 
rod in 8'' . . 


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90 DISCUSSION OF DEFLECTIONS OF THE TWO SLABS 

It will be seen from Tables 1 and 3, that the first three loads 
were practically the same for both slabs. In the Norcross slab 
load 3, of 18 tons, stressed the steel up to the yield point, but in 
the mushroom slab the stress was so small, (being in fact less than 
ten per cent of the former) as probably not to remove all the com- 
pression from the concrete in which it was embedded. Indeed the 
load on the latter slab became five times as much, 90 tons, before 
its steel approached the yield point, at which time it was carrying 
about twice the load which caused the complete failure of the 
Norcross slab. 

Moreover the deflection of the Norcross slab under load 
3, was twenty- two times that of the mushroom slab under the 
same load. This result is in full accord with slab theory which shows 
that the central deflection of a continuous diagonal beam with fixed 
ends uniformly loaded with one sixth of the total load on the slab 
and having the same thickness and reinforcement as the diagonal 
belt, would have more than six times the central deflection of the 
slab, while the stress in its steel would be three or four times as 
much. This gives a measure of the effect of slab action. 

By the phrase ''slab action" we designate the increased strength 
and stiffness of the slab by reason of its resistance to circumferential 
stresses around the colunms and around the center of the panel. 

Furthermore, if this continuous beam be compared with a simple 
beam uniformly loaded and having the same reinforcement, the 
latter would have five times the defiection of the continuous beam, 
or thirty times that of the slab, while the stress in the steel would 
be one and one-half times that in the continuous beam, and six or 
seven times that in the slab. This last exhibits the effect of canti- 
lever action combined with slab action. 

The apparent discrepancy between the observed ratio of de- 
flections in these two slabs of 22 and the just computed deflections 
of 30, is to be accounted for by the fact that the computation 
assumed equal spans, whereas the Norcross span was assumed 
to be diminished from 144'' to 132'' by the column plate. A re- 
duction of the span of this amount will change the computed de- 
flections in the ratio of 144^ : 132^ : : 30 : 23 which is in practical 
agreement with the observed result of 22. 



SUMMARY OF TEST OF MUSHROOM SLAB 91 

By the phrase ^'cantilever action" we designate the increased 
strength and stiffness which is due to the continuity of the beam 
or slab at its supports so that it is convex upwards at such points. 

While the concentration of the loading toward the middle of 
the panel, such as was the case in this test, may prevent any pre- 
cise agreement of these numerical estimates based on uniform 
loading with the results of the tests, they cause the general agree- 
ment shown in the tables and tend strongly to sustain our confi- 
dence in the validity of the analysis from which these concordant 
approximate estimates are obtained. 

The amazing difference in the strength and stiffness of these 
two slabs, which contain practically the same amount of concrete 
and steel, is due to the difference of principle of their construction, 
which may be summarized for the mushroom system by consider- 
ing its slab action and its cantilever action under the follo'wdng 
counts, viz: 

1st. Circumferential slab stresses are most economically and 
effectively provided for by the ring rods around the column heads. 

2nd. The size of the mushroom heads is such as to make the 
belts so wide as to provide reinforcement over the entire area of 
the slab, thus securing slab action in the central part of the panel 
where the belts lie near the lower surface. 

3rd. The reinforcing belts cover a wide zone at the top of 
the slab over the columns and mushroom head, which thus provides 
resistance to tension, and ensures effective cantilever and slab action. 

4th. Concrete is thus stressed in compression at the bottom 
of the slab for a wide zone around the columns. 

5th. Under a load equal to the breaking load of the Norcross 
slab, amounting to thirty-eight tons, the mushroom slab deflected 
at first only l/8'', but after exposure to rain and great changes of 
temperature for seven days had somewhat softened the concrete 
the defi(iction increased to l/4''. 

6th. The first crack appeared underneath the cnlge of the 
slab across the side belt under load No. 5, of fifty-six tons, with a 
center deflection of 0.4'' and an average deflection at (he iniddh^ 
of side belts of 0.25''. 

7th. No cracks appeared on th(^ u])per side of slab nt tlu^ 
edg(», nor wvrv any s(H^n els(nvliere, until h)ad No. 7, of 90 tons was 
apphed, wlien the yield ])()int of tlie stec^l was (^vidtMitly nc^arly or 
(iuit(^ reached, giving a ('(^nfer dc-fUM'tion of 1 2". 



92 



FAILURE OF MUSHROOM SLAB 




Fig. 18. Failure of ^Mushroom Slab. 




Fig. 19. Failure of ISIushroom Slab. Load Removed. 



FAILURE OF MUSHROOM TEST SLAB 93 

8th. The slab carried its final load of over 120 tons for twenty- 
four hours without giving way. It demonstrated the impossibility 
of its sudden failure by gradually yielding until it reached a final 
deflection of some nine inches, as seen in the views of Dec. 17th 
and 24th, Figs. 18 and 19. 

9th. While the slab steel in each belt was the same as in the 
Norcross slab, the crossing of the belts increased the percentage 
of slab reinforcement so much above that of the simple belt rein- 
forcement that stress in the steel did not pass the yield point and 
the failure was largely due to the giving way of the concrete around 
the cap, but partly to some yielding at the line of weakest ultimate 
resistance, both of which statements are confirmed by the view of 
Dec. 24th, Fig. 19, where the removal of the loading permits the 
irregular circular line previously mentioned to be made out at a 
distance from the center of each column of somewhat less than L/2. 

Less steel is required in this system than in the Nor- 
cross slab for the same limiting stresses. Since the steel in this 
slab did not pass the yield point any greater percentage of reinforce- 
ment would be useless and would not increase the strength of the 
slab. It has been found that good practice requires a percentage 
of steel dependent in the following manner upon the thickness 
of the slab: 

li d = L/35 the belt reinforcement = 0.2% 

li d = L/24 the belt reinforcement = 0.3% 

li d = L/20 the belt reinforcement = 0.4% 

Comparision of the steel in the test slabs: Norcross. Mushroom. 

Size of slab 16' x 16' 18.4' x 17.8' 

Area of slab 256 sq. ft. 328 sq. ft. 

Leng-th of 3/8" rods in the slab 1188 ft. 1450 ft. 

Weight of 3/8" rods in the slab 446 lbs. 544 lbs. 

Weight of Plates or Heads in the slab. . . 268 lbs. 435 lbs. 

Total weight of steel in the slab 714 lbs. 979 lbs. 

Weight of steel per square foot of slab.. 2.8 lbs. 3 lbs. 

Area of Panel 12 x 12 ft 144 sq. ft. 144 sc]. ft. 

Length of slab rods per panel ()3S ft. 638 ft. 

Weight of slab rods per panel 239 lbs. 239 li>s. 

Weight in ])lates or heads jkt panel 67 lbs. 109 lbs. 

Total \v(Mght of stec^l ])er j)anel 30() lbs. 3 IS lbs. 

Weight of steel per scjuare foot of ])anel. 2 l/8 lbs. 2 5 12 lbs. 



SUGGESTIONS REGARDING THE CONSTRUCTION 
AND FINISH OF FLOOR SLABS 

By C. A. P. TURNER 

18. The Execution of Work: Construction work of any kind 
involves a great responsibility, not only on the part of the designer, 
but also on the part of those in charge of the work, and that re- 
sponsibility is for the safety of those erecting the work. 

Perhaps the construction of no type of building is so free from 
hazard and risk to the lives of those erecting it as reinforced con- 
crete construction when scientifically designed and intelligently 
executed. 

During the last ten or twelve years, the manufacturers of Port- 
land Cement, have through improvements in methods of manu- 
facture and great reduction in cost, placed this material on the 
market at such reasonable rates that it has given a remarkable 
impetus to the construction of concrete work in all lines. Since, as a 
material of construction, it has but recently come into general use, 
it is not surprising that a large part of the engineering and archi- 
tectural profession have not yet become so familiar T\ith its char- 
acteristics, but that designs lacking in conservatism from a scientific 
standpoint have been frequently made, and this combined with. 
the execution of the work by unskilled contractors, has resulted in a 
number of instances in needless sacrifice of life and large property 
losses, such as a more thorough knowledge and study of the char- 
acteristics of the material should entirely prevent. 

It would be neglect of duty to fail even in this short discussion 
to call attention pointedly to those properties and characteristics 
of concrete which must be kno^sn and appreciated by the engineer 
and constructor in order that he may avoid the serious disasters into 
which those ignorant or forgetful of them have been too frequently 
led. 

The Haedening of Concrete: Concrete may be defined as 
an artificial conglomerate stone in which the coarse aggregate or 
space-filler is held together by the cement matrix. The cement 
should conform to the Standard Specifications for Cement, recom- 
mended by the American Society for Testing Materials. 



HARDENING OF CONCRETE 95 

The contractor and architect should, at least, see to it that the 
cement is finely ground, and that it meets the requirements of the 
boiling test. This last may be readily made by forming pats 
of the cement of 3 J to 4 inches in diameter on a piece of glass, knead- 
ing them thoroughly with just enough moisture to make them plastic, 
so that they will hold their shape without flowing, and taper to a 
thin edge. Store the pats under a moist cloth at a temperature of 
sixty-five to seventy-five degrees Fahr. for a period of 24 hours. 
Then place the pats in a kettle or pan of cold water, and after raising 
the temperature of the water to the boiling point, continue boiling 
for a period of four hours. If the pats do not then show cracks, 
and if they harden without cracking or disintegrating, the con- 
structor may be satisfied that the cement is suitable for use in the 
work. Coarse grinding reduces the sand-carrying capacity of the 
cement, and its consequent efficiency. 

The function assigned to the concrete element in the combina- 
tion of reinforced concrete is to resist compressive stresses in bend- 
ing; but Avhen first mixed the concrete is nothing more than mud, 
and in order for it to become the hard, rigid material necessary to 
fulfill its function in the finished work it must evidentl}^ pass in the 
process of hardening thru all stages and varying degrees of hardness 
from mud and partly cured cement to the final stage of hard, rigid 
material. This curing or hardening being a chemical process, does 
not occur in any fixed period of time, save and except the temper- 
ature conditions are absolutely constant. Hence the time at which 
forms may be safely removed is not to be reckoned by a given number 
of days, but rather it must be determined by the degree of hardness 
attained by the cement. In other words, during warm summer 
weather, concrete may become reasonably well cured in twelve or 
fifteen days. If the weather, however, is rainy and chill}-, it may 
not become cured in a month. In the cold, frosty weather of the 
spring and autumn, unless warm water is used in the mix, the con- 
crete may require two or three months to become thoroughly cured, 
while by heating the mixing water, whenever the temj^erature is 
below 50 (U^grees Fahr., the concrete will harden approximately as 
it does during the more favorable season. 

Concrete which has been chilled by the use of ice cold water, 
or that has become chilled within the first day or two of tlu^ tinu^ it 
is cast, has this peculiarity, that it is very difficult indeinl for \\\c 
most expert to determine when it is in such condition that it will 
retain its shai)e after the removal of tlu^ forms. Ouvv having btHMi 
chilled in the early stages, it goes through consecutive stages of 



96 POURING COXCRETE 

sweating with temperature changes, and during these periods it 
sometimes happens that the concrete diminishes in compressive 
strength, and if the props are removed it sags and gets out of shape. 
Such deformation will generally result in checks and fine cracks, 
though there may not be any serious climinition of the ultimate 
strength. These checks may be prevented as explained above by 
the simple method of heating the mixing water whenever the tem- 
perature has dropped below 50 degrees Fahr. In colder weather, 
that is below the freezing point, not only must the water be heated, 
but as a rule the sand and stone too, also a little salt may be ad- 
vantageously used. The work must then be properly housed and 
kept warm for at least three weeks subsequent to pouring. 

Use of Salt in Cold Weather: We have mentioned the use of 
salt in cold weather. The action of salt is two-fold: It retards the 
setting and thus enables us to use water heated to a higher temper- 
ature than we could use without salt. It also lowers the freezing 
point. Should the concrete then be frozen at the subsequent sweat- 
ing period which occurs with a rise in temperature, the salt retains 
the necessary moisture for crystallization because of its affinity for 
moisture, thus preventing the softened concrete from drjTtig out 
and disintegrating through lack of moisture to enable it to crystalize 
and harden properly. The amount of salt to be used is about a 
cup to the sack of cement "^dth the temperature from 18 to 20 de- 
grees Fahr. If the temperature is below this, increase the amount 
of salt, and when working below zero Fahr., use not less than two 
cups of salt to the bag of cement. 

PorRTXG Concrete: Bad work frequently results from im- 
proper pouring, or casting of the work. In filling the forms, the 
lowest portion of the forms should be filled first. A column should 
be filled from the center and not from the side of the cap. Filling 
from the center will insure a clean smooth face when the forms are 
removed. Filhng from the side vdl\ frequently give a bad surface 
because the mortar ^ill flow into the center of the column through 
the hooping, lea\"ing the coarse aggregate with voids unfilled at the 
outside. As more concrete is then poured in, the voids between the 
core and the out side portion -^ill become filled, and the soft mor- 
tar will not be able to flow back to completely fill the voids between 
the hooping and the casing. Where the spacing of the hooping is 
wide, this is not so important, but it becomes very important where 
the spiral used has close spacing. It is better to cast the coluron and 
mushroom frame complete, continuing to pour the concrete over 
the center of the column so that it always flows from the column 



TEST FOR HARDNESS. LAP 97 

into the Mushroom slab rather than the reverse. All splices must 
be made in a vertical plane, in a beam preferably at the middle of 
the span, and in a slab at a center line of a panel. 

Test for Hardness in Warm Weather: We have pointed 
out that the criterion governing the safe removal of forms is the 
hardness or rigidity of the concrete. A test of hardness in concrete 
not frozen may be made by driving a common eight-penny nail 
into it; the nail should double up before penetrating more than 
half an inch. The concrete should further be hard enough to 
break like stone in knocking off a piece with the hammer. 
Noting the indentation under a blow with the hammer, gives a 
fair idea of its condition to those having experience. 

Subcentering, as provided in the appended specification, is a 
desirable method of preventing deformation, where the use of the 
forms is desired for upper stories before the concrete is fully cured. 

Test for Hardness in Cold Weather: Concrete freshly 
mixed and frozen hard will not only sustain itself but carry a large 
load in addition, until it thaws out and softens, when collapse in 
whole or in part is inevitable. Partly cured concrete if frozen, 
sweats and softens with a rise in temperature, hence in cold weather 
there is danger of mistaking partly cured concrete made rigid by 
frost for thoroughly cured material. In fact the only test that can 
be depended upon with certainty in cold, frosty weather, is to dig out 
a piece of concrete, place a sample on a stove or hot radiator, and 
note whether, as the frost is thawed out of it, it sweats and softens. 
This gives the builder and engineer a perfectly conclusive test of the 
condition of the concrete as to whether it is cured or merely stiffened 
up by frost. 

Lap of Reinforcement over Supports: Thoroughly tying 
the work together by ample lap in the reinforcement is a prime 
requisite for safety in any form or type of construction. This 
general precaution insures toughness, and prevents instantaneous 
collapse, should the workman exercise bad judgment in premature 
removal of forms. 

Responsibility of the Engineer: The steps which it is 
possible for the engineer to take in securing safe construction iwv 
limited in the first place to tlu^ production of a conservative design, 
and one which will present toughness, so that its failun^ uuiUm* over- 
load or under prematun^ n^noval of tl\e forms will be slow and 
gradual. This he can do, and this we belic^ve he is morally bound 
to do. On the other hand, he cannot design reinforced concrete 



98 CRACKS IX CONCRETE 

work which ^\'ill hold its shape without permanent deformation, mi- 
less it is properly supported until the concrete has had time under 
proper conditions to become thoroughly cured. 

Concrete in setting shrinks, and sometimes cracks by reason of 
this shrinkage, particularly when it hardens rapidly, as it does in 
hot weather. This shrinkage sets up certain stresses in the concrete, 
which, combined ^^ith temperature changes, occasionally manifest 
themselves by subsecjuent cracks in the work. Such checks or 
cracks do not of necessity indicate weakness, proA^ding the concrete 
is hard and rigid, since the steel is intended to take the tensile 
stresses and the concrete the compressive. Such checks sometimes 
cause an unwarranted lack of confidence in the safety and stability 
of the work arising from the common lack of famiharity \\dth the 
characteristics of the material. For example, the owner of a frame 
building would never imagine it to be unsafe because he found a few 
season checks in the timber. He is sufficiently familiar "^^ith the 
seasoning of timber to understand how these checks occur, and that 
in most instances they do not mean a loss of strength, since, as the 
timber hardens by thoroughly dr\dng out. it becomes stronger, as a 
rule, to an amount in excess of any slight weakness which might be 
developed by ordinary season cracks or checks. So in concrete, 
when the general public becomes more familiar with its character- 
istics they will regard as far less important than they now do. checks 
which are produced by temperature and shrinkage stresses, or 
possibly by slight uneciual settlement of supports. 

Proper axd Improper ^Iethods of Floor Finish: In con- 
crete work there are a number of small defects which occur through 
failure to properly manipulate the material, for which the designer 
of the engineering part of the work is frequently censured improperly. 
For example, cases have occurred where a good splice was not 
secured owing to the fact that in very hot weather the stone aggre- 
gate became heated in the sun and was not properly cooled do^m 
before mixing the concrete, and so the water dried out too Cj[uickly, 
while the heat in the stone caused the cement to set so rapidly that 
a good splice to the preA^ous work could not be made. 

The worst trouble, however, which has been obserA'ed, is that 
resulting from poor surface finish of fioors. Improper methods in 
common practice are of two dift'erent kinds. One is the attempt 
to fiifish the work approximately at the time it is cast, making the 
surface fiifish integral ^fith the slab. The cfifficulty ^fith this method 
of finishing lies in the fact that as soon as the columns are cast in 
the story above, unequal moisture conditions are produced around 



FLOOR FINISH 99 

the foot of the column owing to the excess of moisture in the column; 
thus the concrete in the surface of the slab around and near the foot 
of the column is expanded by the excess moisture, and it ultimately 
shrinks, and leaves a series of spider web cracks as it dries out. 
This will occur to a greater or less extent depending on the humidity 
of the surrounding atmosphere during the curing or drying out of the 
floor. If the weather is dry these checks will be ver}^ pronounced 
indeed, though they will not be very deep. If it is rainy and damp, 
and the floor is kept soaked all the time, they may be nearly or quite 
lacking. 

Another objection to this method of finish is that unusual pre- 
cautions must be taken to protect the floor before the centering can 
be placed for a storj^ above, and regardless of the method used to 
protect it the floor usually becames scarred and deeply scratched before 
the work is complete, leaving a surface difficult to satisfactorily repair. 

Another method which leads to bad results is the following: 
The rough slab is cast, and the centering removed in due time, the 
slab cleaned and the finish coat applied in a sloppy or plastic form, 
flowed in place, screeded to approximate surface, and then allowed 
to partly set, so that the finishers can get on the floor and trowel it 
down. A floor finished in this manner looks well when the work is 
new. It does not wear well but dusts badly, pits and rapidly grows 
rough and ragged under trucking. 

The correct method of applying floor finish is as follows: 
The finish coat should be not less than 1 and l/4 inches to 1 and l/2 
inches in thickness. It should be applied after the rough slab has been 
fairly well cured. The surface of the rough slab should be thoroughly 
cleaned of dirt and laitance and thoroughly soaked with water. 
Then the floor finish, a mixture preferably of one part of cement to 
one and one-half sand (the sand a silicious sand with grains from 
1/8 inch down, if such can be secured), should be thoroughly mixed 
with just enough water to make an extromel}^ stiff paste, one which 
will hold its form if squeezed in the hand, but one which will not run 
or flow, and will need a fair amount of tamping to bring the moisture 
to the surface. This concrete, so mixed, should be applied to the 
rough slab in blocks of from four to five feet square, first <;r()uting 
the rough slab with a neat cement grout, tlien tami) until (hi> moist- 
ure is brought to the surface, level up and trowel imnuHliatoly. 
The cement finish should not be mixed more rapidly than it can be 
api)lied, so that the cement will not be killed by taking a partial set 
before trowelling, which is what occurs where tlu^ finish is a])pli(Hl 
^I^^PIWj JJ-ii^l l'^^<' workmen wait for it to partly harden hi^l'ore tlu\v can 



100 STRIP FILL FLOORS 

get on it to trowel. A finish applied as just stated will stand severe 
usage and last for several years without showing appreciable evi- 
dence of pitting, dusting, or undue wear. 

The addition of ground iron ore, to the amount of twenty 
pounds to the barrel of cement, appears to improve the finish and 
give it a more pleasing color. 

Checks in cement finish have no relation whatever, as a rule, 
to the strength of the work. They will invariably occur in the 
cement finish where the finish coat is too thin. When it is less than 
1 inch or 3/4 inch at one part of the floor with 1 l/4 inches or 1 l/2 
inches at another, the surface will invariably check and crack badly if 
applied at a sloppy consistency and allowed to partly cure before it 
is polished down. We know of no type of construction where there 
has not been much trouble with finished surfaces in such buildings 
as have come under our observation. But experience has shown us 
that these troubles are needless, and can be avoided by the proper 
handling and application of the finishing coat. 

It is difficult indeed to re-educate those who profess to be 
cement finishers, whose experience has been largely in sidewalk 
finish, or work of that character, to appreciate the necessity for a 
different method of executing work in a building; but when this has 
been accomplished the owner will have the use of a floor flnish free 
from the unpleasant defects above pointed out. 

Strips and Strip Fill for Wood Floors: The proper time 
for the application of the strips and fill is immediately after the 
rough slab has become sufficiently hardened to work upon it, for the 
reason that at this time the strips may be spiked to the partially 
hardened concrete and wedged up or lined up to the desired level 
without difficulty. Then the strip fill can be put in with the same 
rig that is used to cast the floor slab. 

The writer prefers the strip fill of the same mixture as the slab 
except where the loads are so light that increased strength and 
stiffness are of no importance. Then a one to three and one-half, 
four, or even five, mix will answer the purpose. No natural cement or 
hme should be used in the mixture, since when it is used, trouble 
almost invariably follows, caused by its extremely slow hardening 
and its retention of moisture until hardening takes place. This 
moisture frequently swells and expands the flooring to such an 
extent that it springs away from the fastenings, thereby necessi- 
tating the entire relaying of the floors. Conservative practice ac- 
cordingly is to use Portland Cement alone, which will dry out far 
quicker than any natural cement or brown lime. 



APPENDIX 



STANDARD SPECIFICATION FOR REINFORCED 
CONCRETE FLOORS 

By C. A. P. TURNER, Consulting Engineer 
Minneapolis, Minn. 



Reinforcement. Reinforcement shall be of sizes of bars shown on the accom- 
panying plans and details which form a part of this specification. 

All reinforcing metal shall be of medium open hearth or Bessemer steel, 
meeting the requirements of the Manufacturers' Standard Specifications, in 
composition, ultimate strength, ductility and elastic limit, and the required 
bending basis. Hard grade may be used for slab rods only. 

Bending. Bending shall preferably be done cold. If the column rods are 
heated and blacksmith work is done, care must be exercised that the steel is not 
burned in the operation, otherwise it will be condemned by the engineer. 

Cement. Cement shall be of good quality of Portland Cement, of a brand 
which has been upon the market and successfully used for at least four years, 
meeting the requirements of the specification adopted by the American Society 
for Testing Materials. 

The contractor shall give the owner the opportunity to test all cement de- 
livered, and shall furnish the use of testing machine for this purpose. 

The cement shall be delivered in good condition and properly protected 
under suitable cover after delivery on the premises so that it may not be damaged 
by moisture. 

Sand. Sand used in the concrete work shall be clean and coarse, meeting 
the requirements and approval of the engineer and architect. 

Stone. Stone used shall be sound, hard stone, free from lumps of clay and 
other soft unsatisfactory material, or hard smelter slag ma}' be used. In size it 
shall be crushed to pass a 1-inch ring, for slabs and columns, and shall be scrcHMied 
free from dirt and dust. 

Concrete. All concrete shall be mixed in a standard batch machine to the 
consistency of brick mortar, so that it will flow slowly and require on)}' puiUlliug 
around the reinforcement. 

Concn^te shall be thoroughly mixed in the following ])roport.ions: one part 
cement, meeting the requircuuents of the standard specifications; two parts 
clean, coarse sand free from clay, loam or other imi)urit ies; and four i>arts crushed 
stone or clean gravel. 

The concrn^te shall be i)()ured in the low i)ort ions of the forms jirst . That is, 
it shall be jHrnnul diriuttly into the column boxes, beam boxes, etc., before it is 



102 STANDARD SPECIFICATIONS 

poured on the slab. It shall be so placed that it will be forced to flow as little 
as possible to get to the required position, since by flowing, the cement is readily 
separated from the mixture. 

Splices. Splices in beams or slabs are to be made in a vertical plane, prefer- 
ably in the center of the panel or beam. 

Proportions, Each sack of cement shall be considered equivalent to one 
cubic foot in volume, and the mixture of the cement, sand and stone used in the 
concrete shall be proportioned by volume on this basis and as hereinafter specified. 

Concrete for footings, columns, beams and rough slabs throughout shall 
consist of a mixture of one cement, two sand and four of crushed stone. 

For the retaining walls, the concrete mixture shall be one cement, three sand 
and five parts of stone. 

Concrete in which the cement has attained its initial set shall not be used on 
the work. Concrete, however, which has slopped out of the mixer, if cleaned 
up within a short time, not over every half hour, may be put back in the mixer, 
and after being thoroughly mixed again with water may be used on the work. 

Forms. All forms for the reinforced concrete shall be substantially made 
and true to line. Any irregularities due to defective workmanship in this re- 
spect, shall be made good as directed by the architect, by dressing down the 
finished work, or removal and properly replacing it in case that it cannot be 
satisfactorily done. 

A fair quality of lumber, preferably 1x6 square edge fencing shall be used 
for the slab forms. This lumber shall be dressed on the side next to the concrete 
except where plaster is specified by the architect for office finish, in which case 
the rough side of the boarding shall be placed upwards, next to the concrete. 

Column Forms. Column forms shall be made up with plank not less than 
1^ inches thick and stayed at intervals not more than 18 inches vertically be- 
tween bands or straps and shall fit closely at the corner joints, or the forms may 
be made of sheet metal. 

Removal of the Forms. Forms shall not be removed under the most 
favorable conditions, prior to two weeks' time, and under less favorable con- 
ditions where the temperature is lower than 50 ° until the concrete is hard and 
rigid. 

The superintendent will keep in mind the fact that it is not the number of 
days time which has elapsed since placing the concrete which shall determine the 
earliest removal of the forms, but rather how rapidly the concrete has thoroughly 
cured and hardened and that the concrete may be readily stiffened up by cold 
and frost which, when it thaws, will sweat and fail to maintain the desired form. 

Sub=Centering. Where a series of floors are cast one above the other, sub- 
centering of substantial posts about 10 feet centers shall be kept in place until 
there are at least two supporting slabs that are well cured and hard so that the 
concrete may not be overstained in the early stages of hardening. 

Placing and Inspection of Reinforcement. Before Commencing the 
Concrete Work, the reinforcement shall be properly placed and inspected by 
the architect or the engineer representing the owner, and not until after this 
inspection and approval may the work of casting the floor proceed. 

The floor slab rods shall be wired together to hold them in the position as 
shown on the plans. Special attention being given to placing the rods in belts 
of the width of the mushroom frame and fairly uniform spacing, although this is 



STANDARD SPECIFICATIONS 103 

of less importance than keeping to the general distribution through the full width 
of the belts of reinforcement. 

In placing the floor slab rods, all those running from column to column 
directly on one side of a panel shall be placed first, then those running at right 
angles, next all those in one diagonal belt, and then those in the other diagonal. 

Where a belt of slab rods runs parallel to a wall place one rod at bottom on 
forms. Then see that belts normal and diagonally are placed, following 
up with slab rods parallel to the wall on the top of normal and diagonal belts. 

In wiring the rods together it is desirable to use No. 16 soft annealed wire, 
taking a piece, say a yard long, fastening an intersection, then carry the wire 
diagonally to the next intersection, taking a half hitch and proceed until this 
piece is used up and making the end fast. Then start with a new piece and 
proceed as before. 

Two lines of ties, crossing and normal to the intersecting belts at the center 
will hold these rods in position very nicely. 

A similar tie across the parallel belts, and a suitable number of fastenings 
around the mushroom head are required to hold the bars in position. 

Floor Finish. The finish coat on the rough slab shall not be less than 1 inch 
thick, and the rough slab shall be prepared for its reception as follows : 

The slab shall be thoroughly scrubbed with a steel brush and water, and then 
after it has been thoroughly cleaned from dirt and laitance it shall be kept wet 
for at least six hours. The surface shall then be coated with neat cement grout 
and the finish coat applied. 

The finish coat shall consist of a mixture of one cement to one and one-half 
clean, coarse sand. The finish coat shall be mixed with just enough water to 
make a very stiff paste and not enough to make it soft and sloppy. It shall be 
tamped in place and troweled to a smooth finish. 

Mixing the material wet and sloppy renders it necessary to wait until the 
material hardens somewhat before it is possible to polish it down. In allowing 
it to partly harden the finisher is then obliged to break up the surface of partly 
hardened cement which results in a finished surface that will dust badly, pit 
readily and wear rough under subsequent use, so that this method should not 
be employed. 

This finish coat is to be blocked off in squares along the center line of 
columns, and joints shall be made in this coat between panel joints at five to 
six foot intervals. 

Conduits. Before casting the concrete, the concrete contractor shall see 
that the electric contractor has placed the necessary conduits for the wires. These 
shall be kept above the reinforcement wherever they come in the center of a 
panel, the idea being to have these conduit pipes above the steel and dip down 
into the socket at the junction, or to use a special deep socket which would be 
prcfered by the engineer. 

These conduit i)ipes should be carried below the le\'ol of the reinforce- 
ment around thc^ nnisliroom hc;ads where the reinforcement is of necessity near 
the top of the slab. 

Depositing Concrete in Warm Weather. When the concrete is deposited 
in temperatures above 70 ° Fahr., the slab shall be thoroughly wet down twice 

a day for two days after it has been cast. Any prelinunary shrinkage cracks 
which occur on the surface of the slab due to too r:ii>id drying sliall be 
immediately filled with liquid cement grout. 



104 STANDARD SPECIFICATIONS 

Any concrete work indicating that it has not been thoroughly mixed in the 
required proportions shall be dug out and replaced as directed by the engineer 
and architect. 

Placing Concrete in Cold Weather. Where the temperature is below 45 ° 
Fahr., the water shall be heated to a temperature of at least 110 °. Where the 
temperature is below 30 ° Fahr,, artifical heat shall be used to assist in curing 
the concrete, and this must be continued imtil such a time as the slab is thorough- 
ly cured and dry throughout. 

Pouring Concrete. In the mushroom system concrete shall be poured 
over the center of the column until the column is filled. Then the pouring 
shall be continued until the mushroom and mushroom frame is filled up so that 
the concrete will flow from the column toward the center of the slab and not 
from the center of the slab toward the column. In this way solid concrete 
without joints and planes of imperfect bond will be secured around and in the 
vicinity of column heads, where it is most needed. 

Test. No test shall be made until the concrete is thoroughly cured, is dry, 
hard and rigid throughout. Ninety days of good drying weather at a tempera- 
ture above 60 ° Fahr., either natural or artificial, shall be the criterion as to when 
the test of double the working capacity can be reasonably made. 

General. It is the general intent of this specification to require first class 
work in all particulars, and work unsatisfactory to the engineer and architect 
representing the owners shall be made good by the contractor as they direct. 



PRINTED BY 
HEYWOOD 

MINNEAPOLIS 



LIST OF ONE HUNDRED BUILDINGS SELECTED FROM 
MORE THAN A THOUSAND DESIGNED ON THE 

MUSHROOM SYSTEM 

1906 Johnson-Bovey Go's. Bldg Minneapolis, Minn. 

1906 Hoffman Building Milwaukee, Wis. 

1907 Bostwick Braun Bldg Toledo, Ohio 

1907 Lindeke Warner Bldg St. Paul, Minn. 

1907 Hamm Brewery Bldg " " " 

1907 Smythe Building Wichita, Kans. 

1907 Forman Ford Bldg Minneapolis, Minn. 

1907 Grellet Collins Bldg Philadelphia, Pa. 

1907 Parsons Scoville Bldg Evansville, Ind. 

1907 Born Building Chicago, 111. 

1908 South Dakota State Capitol Pierre, S. D. 

1908 Merchants Ice & Cold Storage Bldg Cincinnati, Ohio 

1908 St. Mary's Hospital Kansas City, Mo. 

1908 John Deere Plow Co Omaha, Nebr. 

1908 Minn. State Prison Bldgs ... (6) Stillwater, Minn. 

1908 Ripley Apartments Tacoma, Wash. 

1908 Velie Motor Bldg Moline, 111. 

1908 Park Grant Morris Bldg Fargo, N. D. 

1909 Con P. Curran Bldg St. Louis, Mo. 

1909 Manchester Biscuit Co's. Bldg Fargo, N. D. 

1909 Blue Line Transfer & Storage Bldg Des Moines, la. 

1909 Cutler Hardware Bldg Waterloo, la. 

1909 Mass. Cotton Mills Boston, Mass. 

1909 McMillan Packing Co St. Paul, Minn. 

1909 Vancouver Ice and Cold Storage Co Vancouver, Bldg. 

1909 Omaha Fireproof Storage Bldg Omaha, Nebr. 

1909 J. I. Case Bldg Oklahoma City, Okla. 

1909 Tibbs Hutchings & Co Minneapolis, Minn. 

1909 Snead Mfg. Bldg Louisville, Ky. 

1909 New England Sanitary Bakery Bldg Decatur, III. 

1909 International Harvester Bldg Milwaukee, Wis. 

1909 Congress Candy Co Grand Forks, N. D. 

1910 Y. M. C. A. Bldg Winnipeg, Uim. 

1910 West Publishing Co's. Bldg St. Paul, Minn. 

1910 Beatrice Creamery Bldg Lincoln, Nebr. 

1910 Iten Biscuit Co Omaha, Nebr. 

1910 Turner Moving & Storage Bldg Denver, Colo. 

1910 Congress Realty Co's. Bldg Porthuul, M(^ 

1910 Sniders & Abrahams Bldg Melbourne, Australia 

1910 Strong & Warner Bldg St. Paul, Minn. 

1910 Lexington High School Bldg St. Paul, xMinu. 

1910 Weickcr Transfer & Storage Bldg Denver, C^olo. 

1910 ('h(>hallis County Court House Montesano, \\ash. 

1910 Missouri Glass Co's. Bldg St. Louis, Mo. 

1910 huhistrial Bldg Newark. N. .1. 

1910 Hovel & Wagner Bldg LittU> Kock, .\rk. 

1910 Jobs! Bethanl Bldg IVoria. 111. 

1910 International llarveslcM- (Keystone Works).. SttM-ling, 111. 



1910 Patterson Hotel Bismarck, N. D. 

1910 O'Neil Bldg Akron, Ohio 

1911 Lindsay Bldg Winnipeg, Man. 

1911 King George Hotel Saskatoon, Sask. 

1911 Northern Cold Storage Bldg Duluth, Minn. 

1911 Leighton Supply Co Fort Dodge, la. 

1911 Kinsey Bldg Toledo, Ohio 

1911 Lozier Motor Bldg Detroit, Mich. 

1911 MuUin Warehouse Bldg Cedar Rapids, la. 

1911 Griggs Cooper & Co St. Paul, Minn. 

1911 Swift Canadian Go's. Bldgs Vancouver, B. C. 

1911 McKenzie Bldg Brandon, Man. 

1911 Swift Canadian Go's. Bldg Fort WiUiam, Ont. 

1911 Commerce Bldg St. Paul, Minn. 

1911 Experimental Eng. Bldg. Univ. of Minn Minneapolis, Minn. 

1911 St. Paul Bread Go's. Bldg St. Paul, Minn. 

1911 Rust Parker Martin Bldg Duluth, Minn. 

1912 Woodward Wight Co. Ltd. Bldg New Orleans, La. 

1912 Internationa] Harvester Go's. Bldg .Fort William, Ont. 

1912 H. W. Johns-Manville Bldgs. ... (3) Findeme, N. J. 

1912 Cooledge Bldg Atlanta, Ga. 

1912 Lawrence Leather Go's. Bldg Lawrence, Mass. 

1912 Sears, Roebuck & Co Dallas, Texas 

1912 Vineburg Bldg Montreal, Quebec 

1912 Imperial Tobacco Co Montreal, Quebec 

1912 Richards Pinhorn Bldg Denver, Col. 

1912 Kinney & Levan Co. Bldg Cleveland, Ohio 

1912 Standard Oil Co. Bldgs ... (2) Cleveland, Ohio 

1912 Silver Sunshine Bldgs ... (2) Cleveland, Ohio 

1912 Commercial Improvement Go's. Bldg Columbus, O. 

1912 Moore Department Store Bldg Memphis, Tenn. 

1912 Main Eng. Bldg. Univ. of Minn Minneapohs, Minn. 

1912 Honeyman Hardware Bldg Portland, Ore. 

1912 Revillon Wholesale Hardware Bldg Edmonton, Alta. 

1912 Calgary Furniture Go's. Bldg Calgary, Alta. 

1912 Willoughby Sumner Bldg Saskatoon, Sask. 

1912 U. S. Post Office Minneapolis, Minn. 

1912 Motor Mart Bldg Sioux City, la. 

1912 Finch Van Slyke & McConville Bldg . St. Paul, Minn. 

1912 Hudson Bay Go's. Warehouse Winnipeg, Man. 

1912 Snell Bldg ^ Moose Jaw, Sask. 

1913 Y. M. G. A. Bldg Vancouver, B. C. 

1913 Reynolds Tobacco Factory Bldg Winston Salem, N. C. 

1913 Ford Motor Bldg Memphis, Tenn. 

1913 Ford Motor Bldg Los Angeles, Gal. 

1913 G. Sommers & Co. Bldg St. Paul, Minn. 

1913 Knickerbocker Bldg Los Angeles, Gal. 

1913 Trinity Auditorium Bldg Los Angeles, Gal. 

1913 U. S. Alumium Go's. Bldg Pittsburg, Pa. 

1913 Gordon Fergusen Go's. Bldg St. Paul, Minn. 

1913 S. H. Kress & Go's. Bldg Houston, Tex. 

1913 "Los Muchachos" Bldg San Juan, Porto Rico 



MAY 9 1913 



